COORDINATE ALGEBRAS OF EXTENDED AFFINE LIE ALGEBRAS OF TYPE A_{1}

Yoji Yoshii

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, K1N 6N5 Canada yyoshii@uottawa.ca

Abstract. The cores of extended affine Lie algebras of reduced types were classified except for type A_{1}. In this paper we determine the coordinate algebra of extended affine Lie algebras of type A_{1}. It turns out that such an algebra is a unital ℤ^{n}-graded Jordan algebra of a certain type, called a Jordan torus. We classify Jordan tori and get five types of Jordan tori.

Introduction

Extended affine Lie algebras form a new class of infinite dimensional Lie algebras, which were first introduced by Høegh-Krohn and Torresani in 1990 [7] (under the name of irreducible quasi-simple Lie algebras) as a generalization of the finite dimensional simple Lie algebras and the affine Kac-Moody Lie algebras, and systematically studied in the recent memoir [1].

Roughly speaking, an extended affine Lie algebra, EALA for short, is a complex Lie algebra which has a nondegenerate symmetric invariant form, a self-centralizing finite dimensional ad-diagonalizable abelian subalgebra and a discrete irreducible root system such that elements from non-isotropic root spaces are ad-nilpotent. The core of an EALA is defined as the subalgebra generated by the non-isotropic root spaces.

Typeset by AMS-TEX

2 COORDINATE ALGEBRAS OF TYPE A_{1}

One has a description of an EALA L of type A_{l} (l ≥ 2), D_{l} and E_{l} due to Berman, Gao and Krylyuk [5]. Their description is a 2-step process:

A) describe the core L_{c}, and then, B) describe how L_{c} sits in L.

This program is currently being worked out for the other types of EALA’s. In particular, Allison and Gao [2] describe the cores of all non-simply laced reduced types, i.e., B_{l}, C_{l}, G_{2} and F_{4}. In this paper we describe the cores of the only missing case in reduced types, namely, EALA’s of type A_{1}.

For motivation, we recall the definition of quantum tori and then describe the core of an EALA of type A_{l}, l ≥ 2.

Definition. An n × n matrix q = (q_{ij}) over a field F such that q_{ii} = 1 and q_{ji} = q_{ij}^{−1} is called a quantum matrix. The quantum torus F_{q} = F_{q}[t_{1}^{±1}, …, t_{n}^{±1}] determined by a quantum matrix q is defined as the associative algebra over F with 2n generators t_{1}^{±1}, …, t_{n}^{±1}, and relations t_{i}t_{i}^{−1} = t_{i}^{−1}t_{i} = 1 and t_{j}t_{i} = q_{ij}t_{i}t_{j} for all 1 ≤ i, j ≤ n. Note that F_{q} is commutative if and only if q = 1 where 1 is the quantum matrix whose entries are all 1. In this case, the quantum torus F_{1} becomes the algebra of Laurent polynomials F [t_{1}^{±1}, …, t_{n}^{±1}] in n variables.

A quantum torus is characterized as a unital Λ-graded associative algebra A = ⊕_{α∈Λ} A_{α} over F satisfying

(1) A_{α}A_{β} = A_{α+β} for all α, β ∈ Λ, i.e., A is strongly graded,

(2) dim_{F} A_{α} = 1 for all α ∈ Λ. Note that all nonzero homogeneous elements of A are invertible.

Let

sl_{l+1}(F_{q}) := {X ∈ M_{l+1}(F_{q}) | tr(X) ∈ [F_{q}, F_{q}]}

be the subalgebra of the Lie algebra M_{l+1}(F_{q}) over F of (l + 1) × (l + 1) matrices over F_{q} where tr(X) is the trace of X and [F_{q}, F_{q}] is the span of all commutators [a, b] = ab − ba. It is shown in [5] that the core of any EALA of type A_{l}, l ≥ 3, is a central extension of sl_{l+1}(ℂ_{q}) for F = ℂ, the field of complex numbers.

The central extensions of the Lie algebra sl_{3}(ℂ_{q}) are examples for cores of an EALA of type A_{2}, but do not give all possibilities. Rather, there exists a construction which associates to every alternative algebra A a Lie algebra psl_{3}(A), and it is shown in [6] that a core of an EALA of type A_{2} is a central extension of psl_{3}(A) where A is a unital Λ-graded alternative algebra A = ⊕_{α∈Λ} A_{α} over ℂ satisfying (1) and (2) above. These alternative algebras have been classified in [6]. Besides ℂ_{q}, there exists up to isomorphisms one more type, the Cayley torus 𝕆_{t}. It is defined as 𝕆_{t} = (ℂ[t_{1}^{±1}, …, t_{n}^{±1}], t_{1}, t_{2}, t_{3}), i.e., the octonion algebra over ℂ[t_{1}^{±1}, …, t_{n}^{±1}] obtained by the Cayley-Dickson process with the structure constants t_{1}, t_{2} and t_{3}.

Now, for EALA’s of type A_{1}, the Tits-Kantor-Koecher construction which associates to every Jordan algebra J a Lie algebra TKK(J), called the TKK algebra of J (see e.g. [8]), comes into play. We show:

Theorem 1. The core of any EALA of type A_{1} is a central extension of TKK(J) where J is a unital Λ-graded Jordan algebra J = ⊕_{α∈Λ} J_{α} over ℂ satisfying (T1) {α ∈ Λ | J_{α} ≠ (0)} generates Λ, (T2) all nonzero homogeneous elements are invertible, (T3) dim_{ℂ} J_{α} ≤ 1 for all α ∈ Λ.

Such a graded Jordan algebra over any field F of characteristic ≠ 2 is called a Jordan n-torus or simply a Jordan torus. We classify Jordan tori not only over ℂ but over F. The simplest example of Jordan tori over F is the plus algebra F_{q}^{+} of a quantum torus F_{q}, which is defined on the space F_{q} with the new product x · y := \frac{1}{2}(xy + yx) for x, y ∈ F_{q}. We note that sl_{2}(F_{q}) ≅ TKK(F_{q}^{+}). To state our result, we briefly describe other examples of Jordan n-tori over F.

(a) Let ε = (ε_{ij}) be a quantum matrix such that ε_{ij} = 1 or −1 for all i, j. Define an involution ∗ on F_{ε} = F_{ε}[t_{1}^{±1}, …, t_{n}^{±1}] such that t_{i}^{∗} = t_{i} for all i. Then the symmetric elements H(F_{ε}, ∗) form a Jordan torus. It is a subalgebra of F_{ε}^{+}.

Also, let E be a quadratic field extension of F with the nontrivial Galois automorphism σ_{E}. Let ξ = (ξ_{ij}) be a quantum matrix such that ξ_{ij}σ_{E}(ξ_{ij}) = 1 for all i, j. Define a σ_{E}-semilinear involution σ on E_{ξ} = E_{ξ}[t_{1}^{±1}, …, t_{n}^{±1}] over F such that σ(t_{i}) = t_{i} for all i. Then the symmetric elements H(E_{ξ}, σ) form a Jordan torus over F. It is an F -subalgebra of E_{ξ}^{+}.

(b) Let 2 ≤ m ≤ n and let S^{(m)} be any semilattice in ℤ^{m} (see 1.2 for the precise definition). One can construct a Jordan algebra J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}) over F [t_{1}^{±1}, …, t_{n}^{±1}] of a certain symmetric bilinear form which depends on S^{(m)} and a family of nonzero elements a_{ϵ} ∈ F indexed by some set I (the details are in 5.2). It turns out that J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}) is a Jordan n-torus called a Clifford torus. Clifford tori are a slight generalization of a construction which already appeared in [1].

(c) Suppose that F contains a primitive 3rd root of unity ω. Let ω = (ω_{ij}) be a quantum matrix such that ω_{12} = ω, ω_{21} = ω^{2} and ω_{ij} = 1 for the other i, j. Let 𝔸_{t} = (F_{ω}, t_{3}) be the first Tits construction, using the quantum torus F_{ω} = F_{ω}[t_{1}^{±1}, …, t_{n}^{±1}] and the structure constant t_{3} (details are in 6.8). The central closure of 𝔸_{t} is a 27dimensional exceptional Jordan division algebra over a rational function field in n variables. We will see that 𝔸_{t} is a Jordan torus, which is called the Albert torus. This torus was independently found in [1] and [20]. It is a coordinate algebra of EALA’s of type G_{2} (see [1] and [2]).

We can now state our main result:

Theorem 2. Let J be a Jordan torus over F. Then J is isomorphic to one of the five tori

F_{q}^{+}, H(F_{ε}, ∗), H(E_{ξ}, σ), J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}) or 𝔸_{t}.

Since Jordan tori turn out to be strongly prime, we can use Zelmanov’s Prime Structure Theorem [14] as the first step of our proof. Thus, a Jordan torus is either of Hermitian, Clifford or Albert type. For each type we then determine the possible Jordan tori.

This paper consists of 7 sections: In §1 we give the definition of an EALA and prove Theorem 1 above. In §2 some basic concepts of Jordan algebras are reviewed. In §3 general properties of Jordan tori are described. In §4 we show that a Hermitian torus is graded isomorphic to F_{q}^{+}, H(F_{ε}, ∗) or H(E_{ξ}, σ). In §5 we show that any Clifford torus is graded isomorphic to J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}). In §6 the special Jordan tori of central degree 3 are first determined. Then we prove that any Jordan torus of Albert type is graded isomorphic to the Albert torus 𝔸_{t}. As a corollary, we obtain the classification of Jordan tori of central degree 3, which is used in the classification of cores of EALA’s of type G_{2} in [2]. In §7 the classification of Jordan tori are summarized.

This is part of my Ph.D thesis, written at the University of Ottawa. I would like to thank my supervisor, Professor Erhard Neher, for his suggestions.

§1 Extended affine Lie algebras of type A_{1}

We define extended affine Lie algebras [1]. Let L be a Lie algebra over ℂ (the field of complex numbers). Assume that

(EA1) L has a nondegenerate invariant symmetric bilinear form (·, ·). Here ‘invariant’ means that (·, ·) satisfies ([x, y], z) = (x, [y, z]) for all x, y, z ∈ L.

(EA2) L has a nontrivial finite dimensional self-centralizing ad-diagonalizable abelian subalgebra H.

We will be assuming three further axioms about the triple (L, (·, ·), H). To describe them we need some further notation. Because of (EA2), we have

L = ⊕_{α∈H^{∗}} L_{α} and L_{0} = H

where H^{∗} is the complex dual space of H and for any α ∈ H^{∗} we define L_{α} = {x ∈ L | [h, x] = α(h)x for all h ∈ H}. Let R = {α ∈ H^{∗} | L_{α} ≠ (0)}. R is called the root system of L. Note that since H ≠ (0), we have 0 ∈ R. Also, α, β ∈ R, α+β ≠ 0 ⟹ (L_{α}, L_{β}) = {0}. Thus, −R = R by nondegeneracy. Moreover, (·, ·) is nondegenerate on H. As in the classical theory of finite-dimensional complex semisimple Lie algebras, we can transfer (·, ·) to a form on H^{∗}. Let

R^{×} = {α ∈ R | (α, α) ≠ 0} and R^{0} = {α ∈ R | (α, α) = 0}.

The elements of R^{×} (resp. R^{0}) are called non-isotropic (resp. isotropic) roots. We have R = R^{×} ∪ R^{0}. We further require that

(EA3) α ∈ R^{×}, x_{α} ∈ L_{α} ⟹ ad x_{α} acts locally nilpotently on L.

(EA4) R is a discrete subset of H^{∗}.

(EA5) R is irreducible. That is,

(a) R^{×} = R_{1} ∪ R_{2}, (R_{1}, R_{2}) = (0) ⟹ R_{1} = ∅ or R_{2} = ∅

(b) σ ∈ R^{0} ⟹ there exists α ∈ R^{×} such that α + σ ∈ R^{×}. If L satisfies (EA1)-(EA5), the triple (L, (·, ·), H), or simply the algebra L itself, is called an extended affine Lie algebra or EALA for short.

Let t_{α} be the unique element of H so that (t_{α}, h) = α(h) for h ∈ H, and put h_{α} = \frac{2}{(α,α)}t_{α}. Then there exist nonzero e_{α} ∈ L_{α} and f_{α} ∈ L_{−α} so that [h_{α}, e_{α}] = 2e_{α}, [h_{α}, f_{α}] = −2f_{α} and [e_{α}, f_{α}] = h_{α}. In other words, {e_{α}, h_{α}, f_{α}} is an sl_{2}-triplet. Thus we can use sl_{2}-theory. Assuming only (EA1), (EA2) and (EA3), one can show that some well-known properties of finite dimensional semisimple Lie algebras over ℂ are also true for EALA’s.

1.1. ([1], Lemma I.1.21 and Theorem I.1.29) Let α ∈ R^{×}, β ∈ R and r = 2\frac{(β,α)}{(α,α)}. Then:

(i) 2\frac{(β,α)}{(α,α)} ∈ ℤ.

(ii) dim_{ℂ} L_{α} = 1.

(iii) Assume that ade_{α}(e_{β}) = 0. Then r ≥ 0,

(adf_{α})^{i}(e_{β}) ≠ 0, for all i = 0, 1, …, r, and (adf_{α})^{r+1}(e_{β}) = 0. □

In the following, L is an EALA with root system R. We recall some of the properties of R that we will need. First, note that if a nondegenerate invariant symmetric bilinear form on L is multiplied by a nonzero complex number, then we still have such a form. Since the axioms are invariant under such a change, we may as well assume that there is some non-isotropic root α ∈ R^{×} with (α, α) ∈ ℝ_{>0}. Then if β ∈ R^{×} we have 2\frac{(β,α)}{(α,α)} ∈ ℤ so that (β, α) ∈ ℝ, and hence, since 2\frac{(α,β)}{(β,β)} ∈ ℤ we get (β, β) ∈ ℝ if (α, β) ≠ 0. It now follows, using (EA5)(a), that (α, β) ∈ ℝ for any α, β ∈ R. That is, our form is real valued on the real linear span of the roots. From now on we assume that our form is scaled so that there is at least one α ∈ R^{×} with (α, α) > 0. Let V be the real span of R in H. Then it was proven in [1] Theorem I.2.14 that the real valued symmetric bilinear form (·, ·) |_{V} is positive semidefinite on V. This was a conjecture of Kac. Let

V^{0} = {α ∈ V | (α, β) = 0 for all β ∈ V}.

The nullity of R or of L is defined to be the real dimension n of V^{0}. Let \overline{V} = V/V^{0} and let : V ⟶ \overline{V} be the canonical projection. Then (·, ·) induces a positive definite symmetric bilinear form on \overline{V} so that, relative to this form, the image \overline{R} of R in \overline{V} is a finite irreducible (possibly non-reduced) root system [1] Proposition I.2.19. The type of R or of L is defined to be the type of the finite root system \overline{R}.

Our interest is in EALA’s of type A_{1}. Thus \overline{R} = \{\overline{0}, ±\overline{α}\}. We choose a fixed preimage α̇ ∈ R of \overline{α} under . Let V̇ be the subspace spanned by α̇ of V. Then we have V = V̇ ⊕ V^{0}, and restricts to an isometry of V̇ onto \overline{V}. Let Ṙ be the image of R under the projection of V onto V̇, and so Ṙ = {0, ±α̇}. We define

S = {σ ∈ V^{0} | α̇ + σ ∈ R} and Λ = the subgroup of V^{0} generated by R^{0}.

Then, by [1] Chapter II, we have R^{0} = S + S and R = (S + S) ∪ (α̇ + S) ∪ (−α̇ + S). Moreover, S is a discrete spanning set in V^{0}, Λ is a lattice in V^{0} and S is a semilattice in Λ:

Definition 1.2. A subset S of a lattice Λ which has the following three properties, (i) 0 ∈ S, (ii) 2σ − τ ∈ S for all σ, τ ∈ S, (iii) S generates Λ,

is called a semilattice in Λ.

We put

L^{σ} := L_{−α̇+σ} ⊕ L_{σ} ⊕ L_{α̇+σ} for σ ∈ Λ.

Then we have L = ⊕_{σ∈Λ} L^{σ} and [L^{σ}, L^{τ}] ⊂ L^{σ+τ} for σ, τ ∈ Λ. In other words, L is a Λ-graded Lie algebra. Definition 1.3. The core of L is defined to be the subalgebra L_{c} of L generated by the spaces L_{α}, α ∈ R^{×}.

Since L_{c} is generated by homogeneous elements, L_{c} is a Λ-graded subalgebra of L. Thus, L_{c} = ⊕_{σ∈Λ} (L_{c})^{σ}, where for σ ∈ Λ, and

(L_{c})^{σ} := L_{c} ∩ L^{σ} = L_{−α̇+σ} ⊕ ∑_{τ,ν∈Λ,τ+ν=σ}[L_{α̇+τ} , L_{−α̇+ν}] ⊕ L_{α̇+σ}.

Also, let

(L_{c})_{−α̇} := ⊕_{σ∈Λ} L_{−α̇+σ}, (L_{c})α̇ := ⊕_{σ∈Λ} L_{α̇+σ}, (L_{c})_{0} := ∑_{τ,ν∈Λ} [L_{α̇+τ} , L_{−α̇+ν}].

Then we have L_{c} = (L_{c})_{−α̇} ⊕ (L_{c})_{0} ⊕ (L_{c})α̇. Let Ġ := ⟨eα̇, hα̇, fα̇⟩ ≅ sl_{2}(ℂ) and Ḣ := ℂhα̇, which are both subalgebras of L_{c}. Obviously, for each ε = 0, −1, 1

(L_{c})_{εα̇} = {x ∈ L_{c} | [h, x] = εα̇(h)x for all h ∈ Ḣ},

and L_{c} is generated as an algebra by (L_{c})_{−α̇} and (L_{c})α̇. Therefore, L_{c} is graded by the root system of type A_{1} as defined in [4]. By the description of such Lie algebras (see [3] or [15]), L_{c} is a central extension of the TKK algebra of a unital Jordan algebra. The Jordan algebra J, called the coordinate algebra of L of type A_{1}, is defined as follows. Let J := (L_{c})α̇ as a ℂ- vector space and define a multiplication on J by

xy := \frac{1}{2}[[x, fα̇], y] for x, y ∈ J.

One can check that this multiplication is commutative and satisfies the Jordan identity, and so J is a Jordan algebra over ℂ. Note that eα̇ is the identity element of J. Our goal is to describe the structure of J. We put J_{σ} := L_{α̇+σ}. Then 1 := eα̇ ∈ J_{0}, and one can easily see that J = ⊕_{σ∈Λ} J_{σ} is a Λ-graded Jordan algebra over ℂ, i.e., J_{σ}J_{τ} ⊂ J_{σ+τ} for all σ, τ ∈ Λ. Also, by 1.1(ii), we have

dim_{ℂ} J_{σ} = 1 if σ ∈ S and J_{σ} = (0) if σ ∈ Λ \ S.

These conditions are not enough to classify J. To obtain a crucial property of J, ‘invertibility of nonzero homogeneous elements’, we use the following well-known identity in the theory of Jordan pairs [15]:

(1.4)  U_{x}y = \frac{1}{2}[x, [x, ȳ]] where ȳ:= \frac{1}{2}[fα̇, [fα̇, y]] for all x, y ∈ J,

where the U-operator U_{x} will be defined in (2.0). Also, we need the following easy consequence of sl_{2}-theory:

Lemma 1.5. For σ, τ ∈ S and ε = ±1, we have

[e_{εα̇+σ}, [e_{εα̇+σ}, e_{−εα̇+τ}]] ≠ 0,

and hence, [L_{εα̇+σ}, [L_{εα̇+σ}, L_{−εα̇+τ}]] = L_{εα̇+2σ+τ}.

Proof. We only show the case ε = 1 since the case ε = −1 is done by the same manner. Take α := −α̇−σ and β := −α̇+τ in 1.1(iii). Then [e_{α}, e_{β}] ∈ L_{−2α̇−σ+τ} = (0) since A_{1} is reduced. Also, we have 2\frac{(β,α)}{(α,α)} = 2, and hence by 1.1(iii), [f_{−α̇−σ}, [f_{−α̇−σ}, e_{−α̇+τ}]] ≠ 0. Since dim_{ℂ} L_{α̇+σ} = 1 (see 1.1(ii)), there exists some 0 ≠ c ∈ ℂ such that f_{−α̇−σ} = ce_{α̇+σ}, and so 0 ≠ [e_{α̇+σ}, [e_{α̇+σ}, e_{−α̇+τ}]] ∈ L_{εα̇+2σ+τ}. □

For σ ∈ S, let 0 ≠ x ∈ J_{σ}. Since S is a semilattice, there exists 0 ≠ y ∈ J_{−2σ}. Then, by 1.4 and 1.5, we have 0 ≠ \frac{1}{2}[x, [x, ȳ]] = U_{x}y ∈ J_{2σ−2σ} = J_{0}. Since dim_{ℂ} J_{−2σ} = dim_{ℂ} J_{0} = 1 and 1 ∈ J_{0}, there exists c ∈ ℂ such that U_{x}(cy) = 1. Hence, x is invertible (see 2.2(2)). Thus any nonzero element in J_{σ} for all σ ∈ S is invertible. Consequently, we have obtained some necessary conditions of the coordinate algebra J, namely,

Theorem 1.6. The core of an EALA of type A_{1} is isomorphic to a central extension of the TKK algebra of a unital Λ-graded Jordan algebra J = ⊕_{σ∈Λ} J_{σ} over ℂ satisfying (T1) {σ ∈ Λ | J_{σ} ≠ (0)} generates Λ, (T2) all nonzero homogeneous elements are invertible, (T3) dim_{ℂ} J_{σ} ≤ 1 for all σ ∈ Λ. □

We will classify such Jordan algebras not only over ℂ but over any field F of ch. F ≠ 2 in later sections. By the argument above, we know that 1 ∈ J_{0} and that {σ ∈ Λ | J_{σ} ≠ (0)} is a semilattice in Λ. However, we do not need to assume the properties since such Jordan algebras already satisfy them (see 3.5).

§2 Review of Jordan algebras

Throughout F is a field of characteristic ≠ 2. An algebra over F is a “linear” nonassociative algebra A defined as a vector space over F with an F -bilinear map A × A ⟶ A, called multiplication. We assume that an algebra is unital in the sense that there exists 1 ∈ A, called an identity element, such that 1x = x = x1 for all x ∈ A. For an algebra A and x, y, z ∈ A we define the commutator [x, y] = xy − yx and the associator (x, y, z) = (xy)z − x(yz).

An algebra J over F satisfying the following two identities is called a (linear) Jordan algebra over F: for all x, y ∈ J,

[x, y] = 0 (commutativity) and (x, y, x^{2}) = 0 (Jordan identity).

Let us define the so-called U-operator for x ∈ J, i.e., U_{x} : J ⟶ J by

(2.0)  U_{x}y = U_{x}(y) = 2x(xy) − x^{2}y for all y ∈ J.

The plus algebra A^{+} of an associative algebra A over F is an example of a Jordan algebra: for x, y ∈ A^{+} = (A, ·), with a new multiplication · on A defined as x · y := \frac{1}{2}(xy + yx). For this example, the U-operator is given by U_{x}y = xyx. A Jordan algebra is called special if it is isomorphic to a subalgebra of the plus algebra of some associative algebra. A Jordan algebra is called exceptional if it is not special.

Remark 2.1. It is well known that Jordan algebras are power associative, i.e., the subalgebra generated by any element is associative (and commutative) (see e.g. [21] p.37, p.68).

An element x in a Jordan algebra J is called invertible if there exists y ∈ J such that xy = 1 and x^{2}y = x. In this case y is unique and is denoted by x^{−1}.

We denote the subset of invertible elements of an algebra A by A^{×}.

2.2. ([21] p.303-4) Let A be an associative algebra and J a Jordan algebra. Then:

(1) A^{×} = (A^{+})^{×}, and for x ∈ A^{×} = (A^{+})^{×}, x^{−1} in the associative algebra A and the Jordan algebra A^{+} coincide.

(2) For x ∈ J, x ∈ J^{×} ⟺ U_{x} is invertible ⟺ there exists y ∈ J such that U_{x}y = 1. In these cases, we have U_{x}^{−1} = U_{x^{−1}} and y = x^{−2}.

(3) For x, y ∈ J, x, y ∈ J^{×} ⟺ U_{x}y ∈ J^{×}. In particular, x ∈ J^{×} ⟺ x^{n} ∈ J^{×} for all n ∈ ℤ.

(4) For any x ∈ J^{×}, the subalgebra of J generated by x and x^{−1} is associative (and commutative). □

We recall some basic notions for Jordan algebras.

Definition 2.3. Let J be a Jordan algebra. Then J is called (i) a Jordan domain if U_{x}y = 0 implies x = 0 or y = 0 for all x, y ∈ J, (ii) nondegenerate if U_{x} = 0 implies x = 0 for all x ∈ J, (iii) prime if U_{I}K = (0) implies I = (0) or K = (0) for all ideals I, K of J where

U_{I}K = {∑_{x,y} U_{x}y | x ∈ I, y ∈ K}, (iv) strongly prime if J is nondegenerate and prime.

The nondegeneracy and primeness generalize the notion of a domain, namely,

Lemma 2.4. A Jordan domain has no nilpotents and is strongly prime.

Proof. Straightforward. □

The centre of a Jordan algebra J is defined as

Z(J) = {z ∈ J | (z, x, y) = 0 for all x, y ∈ J}.

We note that if z ∈ Z(J) is invertible, then z^{−1} ∈ Z(J). The following lemma is well-known (see [13] Corollary 3.4, p.12):

2.5. Let A be a semiprime associative algebra and Z(A) its centre. Then we have Z(A) = Z(A^{+}). □

For a prime associative or Jordan algebra A, any 0 ≠ x ∈ Z := Z(A) has no torsion element in A, i.e., xy = 0 for some y ∈ A ⟹ y = 0 (see [9] Proposition 7.6.5, p.7.24). In particular, Z is an integral domain. Thus we can define the tensor algebra \overline{A} = \overline{Z} ⊗_{Z} A over \overline{Z} where \overline{Z} is the field of fractions of Z, and call it the central closure of A. The following lemma is well-known (see e.g. [21] p.186):

2.6. Let A be a prime associative or Jordan algebra and Z = Z(A) its centre. Then we have:

(i) A embeds into \overline{A} via x ↦ 1 ⊗ x for all x ∈ A,

(ii) \overline{A} is a central over \overline{Z}, i.e., Z(\overline{A}) = \overline{Z}.1,

(iii) A is an associative (resp. a Jordan) domain ⟺ \overline{A} is an associative (resp. a Jordan) domain. □

We mention some well-known identities on A^{+} for any associative algebra A. Define x ∘ y = xy + yx for x, y ∈ A and let (·, ·, ·)^{∘} be the associator of this circle product. Then the following identities which can be easily verified by expanding both sides hold: for all x, y, z ∈ A,

(2.7)  [x, [y, z]] = (y, x, z)^{∘},
(2.8)  [x, y]^{2} = x ∘ U_{y}x − U_{x}y^{2} − U_{y}x^{2},
(2.9)  U_{[x,y]} = U_{x∘y} − 2U_{x}U_{y} − 2U_{y}U_{x}.

By Wedderburn’s Structure Theorem, a finite dimensional associative domain is a division algebra. Also, the following is well-known (see [8] p.156 Theorem 2).

2.10. A finite dimensional Jordan domain is a division algebra. □

Finally, there is a notion of degree for finite dimensional simple Jordan algebras (see e.g. [8], p.209), and the degree coincides with the generic degree for them (see [8], p.233). The following lemma seems to be known to the experts, but for the convenience of the reader we include a proof.

2.11. Let J be a finite dimensional central special Jordan division algebra over F of degree r. Then:

(a) r ≠ 2^{m} for m ≥ 1 ⟹ dim_{F} J = r^{2},

(b) r = 3 ⟺ dim_{F} J = 9.

Proof. (a): It is clear for r = 1, and so we assume that r > 1. If J is a finite dimensional central special Jordan division algebra over F of degree r ≠ 2, then J ≅ D^{+} or H(D, ∗) where D is a central associative division over the centre of degree r and ∗ is an involution of D (see Theorem 11 and Exercise 1 in [8] p.210). Thus, if J ≅ D^{+}, then we have dim_{F} J = r^{2}. If ∗ is of the second kind, we know dim_{F} H(D, ∗) = r^{2} (see [10] p.190). If J ≅ H(D, ∗) and r ≠ 2^{m} for m ≥ 1, then there does not exist an involution of first kind on D. For, if one exists, then D ≅ D^{op} (the opposite algebra), and so the order of D in the Brauer group of F is 2. Since any prime factor of the degree of D divides the order (see Theorem 2.7.5 [10] p.61), the degree of D has to be a power of 2. This is a contradiction. Hence (a) has been shown.

(b): By (a), we get r = 3 ⟹ dim_{F} J = 9. Suppose that dim_{F} J = 9. From the classification of the finite dimensional simple associative algebras with involution (see [10] p.190), we have dim_{F} J = r^{2} (J ≅ D^{+} or ∗ is of the second kind), r(r + 1)/2 (∗ is orthogonal) or r(r − 1)/2 (∗ is symplectic). Since r(r + 1)/2 or r(r − 1)/2 is never 9, we get 9 = r^{2}, i.e., r = 3. □

§3 General properties of Jordan tori

Whenever a class of algebras has a notion of invertibility, one can make the following definition:

Definition 3.1. Let G be a group. A G-graded algebra T = ⊕_{g∈G} T_{g} over F satisfying

(T1) supp T := {g ∈ G | T_{g} ≠ (0)} generates G, (T2) all nonzero homogeneous elements are invertible, (T3) dim_{F} T_{g} ≤ 1 for all g ∈ G,

is called a G-torus. Moreover, if

(St) T is strongly graded, i.e., T_{g}T_{h} = T_{gh} for all g, h ∈ G,

then T is called a G-torus of strong type.

When G = Λ is a free abelian group of rank n, T is called an n-torus, or simply a torus. If T is associative or Jordan, it is called an associative torus or a Jordan torus, respectively.

One can easily check that if T = ⊕_{g∈G} T_{g} is a G-graded associative algebra satisfying (T3), then (T1) and (T2) are equivalent to (St). Thus the notions of a G-torus and a G-torus of strong type coincide for the class of associative algebras. Note for a G-torus T = ⊕_{g∈G} T_{g} of strong type, we have dim_{F} T_{g} = 1 for all g ∈ G and supp T = G. We give examples of associative tori and Jordan tori.

From now on, Λ denotes a free abelian group of rank n.

Example 3.2. Let {σ_{1}, …, σ_{n}} be a basis of Λ. We give a Λ-grading to a quantum torus F_{q} = F_{q}[t_{1}^{±1}, …, t_{n}^{±1}] (see the definition in Introduction) in the following way: Define the degree of t_{α} := t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}} for α = α_{1}σ_{1} + ⋯ + α_{n}σ_{n} ∈ Λ to be α. Then this grading makes F_{q} = ⊕_{α∈Λ} F t_{α} into a torus, and we call the grading a toral Λgrading of F_{q}. If one needs to specify a basis of Λ, we call it a ⟨σ_{1}, …, σ_{n}⟩-grading of F_{q}. Any associative torus is graded isomorphic to some F_{q} with some toral grading (see [5] and [6]). Also, any commutative associative torus is graded isomorphic to F_{1} = F [t_{1}^{±1}, …, t_{n}^{±1}], the algebra of Laurent polynomials over F. One can check that the multiplication rule of F_{q} for q = (q_{ij}) is the following: for β = β_{1}σ_{1}+⋯+β_{n}σ_{n} ∈ Λ,

(3.3)  t_{α}t_{β} = ∏_{i<j} q_{ij}^{α_{j}β_{i}}t_{α+β}.

Clearly, F_{q}^{+} with the same grading as F_{q} becomes a Jordan torus and the multiplication rule is the following:

t_{α} · t_{β} = \frac{1}{2}(∏_{i<j} q_{ij}^{α_{j}β_{i}} + ∏_{i<j} q_{ij}^{β_{j}α_{i}}) t_{1}^{α_{1}+β_{1}} ⋯ t_{n}^{α_{n}+β_{n}}

= \frac{1}{2} ∏_{i<j} q_{ij}^{α_{j}β_{i}}(1 + ∏_{i,j} q_{ij}^{α_{i}β_{j}}) t_{α+β}. (3.4)

We call the grading of F_{q}^{+} induced from a toral grading of F_{q} a toral grading of F_{q}^{+}. Note that supp F_{q}^{+} = Λ, and F_{q}^{+} is of strong type if and only if

∏_{i,j} q_{ij}^{α_{i}β_{j}} ≠ −1 for all α, β ∈ Λ.

Unlike the situation for associative tori, we may have supp J ≠ Λ for a Jordan torus J in general. We will give such examples in §4 and §5. We show that supp J cannot be any subset of Λ, namely,

Lemma 3.5. Let J = ⊕_{α∈Λ} J_{α} be a Jordan torus. Then 1 ∈ J_{0} and supp J is a semilattice in Λ.

Proof. In general, we have:

Claim. Let A = ⊕_{g∈G} A_{g} be a G-graded algebra. Then 1 ∈ A_{e} where e is the identity element of G.

Proof. Let 1 = ∑_{g∈G} x_{g} ∈ A. For any u ∈ A_{h}, h ∈ G, we have u = 1u = ∑_{g∈G} x_{g}u ∈ A_{h} since 1 is a left identity element. Since G is a group, we have x_{e}u = u (and x_{g}u = 0 if g ≠ e). Thus x_{e} is a left identity element. Hence we have 1 = x_{e}1 = x_{e} ∈ A_{e} since 1 is a right identity element. □

By this claim, we have 1 ∈ J_{0}. In particular, 0 ∈ supp J. Let 0 ≠ x ∈ J_{α} for α ∈ supp J. Let x^{−1} = ∑_{β} y_{β}. Then ∑_{β} xy_{β} = 1 ∈ J_{0} and ∑_{β} x^{2}y_{β} = x ∈ J_{α} imply that xy_{−α} = 1 and x^{2}y_{−α} = x (xy_{γ} = 0 and x^{2}y_{γ} = 0 for all γ ≠ −α). Thus, by the uniqueness of the inverse, we get x^{−1} = y_{−α} ∈ J_{−α}, and so −α ∈ supp J. For any α, β ∈ supp J, let 0 ≠ u ∈ J_{α} and 0 ≠ v ∈ J_{−β}. Then, by 2.2(3), 0 ≠ U_{u}v = 2u(uv) − u^{2}v ∈ J_{2α−β}, and so 2α − β ∈ supp J. Since supp J generates Λ, supp J is a semilattice in Λ. □

Remark. Any Jordan torus J = ⊕_{α∈Λ} J_{α} satisfies U_{J_{α}}J_{β} = J_{2α+β} for all α, β ∈ supp J.

We recall that Λ is a totally ordered abelian group using the lexicographic order. Thus we have:

Lemma 3.6. (i) A Jordan or an associative torus is a Jordan or an associative domain.

(ii) Any invertible element of a Jordan or an associative torus is homogeneous.

Proof. Both statements are well-known for the associative case (see [11] p.95), and so we only show the Jordan case. Let J = ⊕_{α∈Λ} J_{α} be a Jordan torus. For 0 ≠ x, y ∈ J, suppose that U_{x}y = 0. Let x = x_{α_{0}} + (terms of higher degree) and y = y_{β_{0}} + (terms of higher degree). Then U_{x_{α_{0}}} y_{β_{0}} is the least homogeneous component of U_{x}y, and so U_{x_{α_{0}}} y_{β_{0}} = 0. This is a contradiction since x_{α_{0}} and y_{β_{0}} are invertible (see 2.2(3)). Hence (i) is settled.

For (ii), suppose that x ∈ J is invertible but not homogeneous, i.e., x = x_{α_{0}} + (terms of middle degree) + x_{α_{1}} where α_{0} is the minimum degree and α_{1} is the maximum degree of x with α_{0} < α_{1}. By 2.2(2), there exists y ∈ J such that U_{x}y = 1. Let y = y_{β_{0}} + (terms of middle degree) + y_{β_{1}} where β_{0} is the minimum degree and β_{1} is the maximum degree of y (could be β_{0} = β_{1}). Then U_{x_{α_{0}}} y_{β_{0}} ∈ J_{2α_{0}+β_{0}} is the minimum degree and U_{x_{α_{1}}} y_{β_{1}} ∈ J_{2α_{1}+β_{1}} is the maximum degree of U_{x}y, and 2α_{0} + β_{0} < 2α_{1} + β_{1}. This contradicts the fact that U_{x}y = 1 is homogeneous. □

Corollary 3.7. Let T = ⊕_{α∈Λ} T_{α} be a Jordan or an associative torus. Suppose that 0 ≠ x ∈ T and x^{m} ∈ T_{β} for some β ∈ Λ. Then we have β ∈ mΛ and x ∈ T_{\frac{1}{m} β}.

Proof. By 3.6(i) and 2.4, we have x^{m} ≠ 0. Since x^{m} ∈ T_{β}, x^{m} is invertible, whence x is invertible (see 2.2(3)). Therefore, by 3.6(ii), we have x ∈ T_{γ} for some γ ∈ Λ, and so mγ = β. □

Let T = ⊕_{α∈Λ} T_{α} be a Jordan or an associative torus. Then the centre Z = Z(T) of T is a homogeneous subalgebra. Also, it is clear that Z is graded by the subgroup Γ := {γ ∈ Λ | T_{γ} ∩ Z ≠ (0)} of Λ, and so Z = ⊕_{γ∈Γ} T_{γ} is a commutative associative Γ-torus, which is the algebra of Laurent polynomials. We call this Γ the central grading group of T.

Lemma 3.8. Let T = ⊕_{α∈Λ} T_{α} be a Jordan or an associative torus over F with centre Z. Let K be a field extension of F. Then the scalar extension T_{K} = K ⊗_{F} J is a Jordan or an associative torus over K with centre K ⊗_{F} Z, and the central grading groups of T and T_{K} coincide.

Proof. Straightforward. □

For a Jordan or an associative torus T, the central closure \overline{T} = \overline{Z} ⊗_{Z} T over \overline{Z} makes sense (see 3.6(i)).

Lemma 3.9. Let T = ⊕_{α∈Λ} T_{α} be a Jordan or an associative torus with its centre Z, Γ the central grading group of T and \overline{T} the central closure. For \overline{α} ∈ Λ/Γ, let T\overline{α} := ZT_{α} and \overline{T} \overline{α} := \overline{Z} ⊗_{Z} ZT_{α}. Then:

(i) T\overline{α} = T\overline{α′} for all α′ ∈ \overline{α}, and T\overline{α} is a free Z-module of rank 1 if α ∈ supp T and rank 0 otherwise.

(ii) T = ⊕_{\overline{α}∈Λ/Γ} T\overline{α}, which is a free Z-module and a Λ/Γ-graded algebra over Z with rank T\overline{α} ≤ 1 for all \overline{α} ∈ Λ/Γ.

(iii) \overline{T} = ⊕_{\overline{α}∈Λ/Γ} \overline{T} \overline{α}, which is a Λ/Γ-torus over \overline{Z} with dim\overline{Z} \overline{T} = | supp T/Γ|. (iv) The quotient group Λ/Γ cannot be a nontrivial cyclic group.

Proof. (i) is trivial. For (ii), we note that for all γ ∈ Γ, T_{α+γ} = T_{γ}T_{α} since T_{γ} ⊂ Z. Hence we have

T = ⊕_{α∈Λ} T_{α} = ⊕_{\overline{α}∈Λ/Γ} ( ⊕_{γ∈Γ} T_{α+γ}) = ⊕_{\overline{α}∈Λ/Γ} ZT_{α} = ⊕_{\overline{α}∈Λ/Γ} T\overline{α}.

The rest of statements follows from (i). (iii) follows from (ii). For (iv), suppose that Λ/Γ is a nontrivial cyclic group. One can easily check that any Jordan or associative G-torus for a cyclic group G is commutative and associative. Hence, by (iii), \overline{T} is commutative and associative. By 2.6(i), T embeds into \overline{T}, and so Z = T and Γ = Λ, i.e., Λ/Γ is the trivial group. Thus we get a contradiction. □

We will start to classify Jordan tori in the next section. For this purpose we state Zelmanov’s Prime Structure Theorem ([14] p.200) in a short form, designed for our needs. Namely, a strongly prime Jordan algebra J is one of the following three types: (The new terminology used below will be explained in the following sections.)

Hermitian type: J is special and q_{48}(J) ≠ {0},

Clifford type: the central closure \overline{J} is a simple Jordan algebra over \overline{Z} of a symmetric bilinear form,

Albert type: the central closure \overline{J} is an Albert algebra over \overline{Z}.

Since Jordan tori are strongly prime (see 3.6(i) and 2.4), the type of Jordan tori is defined as above.

§4 Hermitian type

We review the so-called Zelmanov polynomial q_{48}. Let FSJ (X) be a free special Jordan algebra on an infinite set X of variables over F. That is, FSJ (X) is the subalgebra of FA(X)^{+} generated by X where FA(X) is the free associative algebra on X. For x, y, z, w ∈ X, let

p_{16}(x, y, z, w) = [[D_{x,y}^{2}(z)^{2}, D_{x,y}(w)], D_{x,y}(w)] ∈ FA(X)

where D_{x,y}(z) = [[x, y], z]. For 12 variables x_{i}, y_{i}, z_{i}, w_{i} ∈ X, i = 1, 2, 3, let

q_{48} = [[p_{16}(x_{1}, y_{1}, z_{1}, w_{1}), p_{16}(x_{2}, y_{2}, z_{2}, w_{2})], p_{16}(x_{3}, y_{3}, z_{3}, w_{3})] ∈ FA(X).

By 2.7, we have p_{16}, q_{48} ∈ FSJ (X). Moreover, q_{48} is homogeneous in each variable, i.e., all monomials of q_{48}, the monomials not only of the associative product but also of the Jordan product, have the same partial degree in each variable. Note that the total degree in 12 variables is 48. For any Jordan algebra J, we denote the evaluation of q_{48} on J by q_{48}(J).

An ideal I ◁ FSJ (X) is called formal if for all permutations σ of X,

p(x_{1}, …, x_{n}) ∈ I ⟹ p(σ(x_{1}), …, σ(x_{n})) ∈ I.

For a formal ideal I of FSJ (X) and any special Jordan algebra J, it is well-known that the evaluation I(J) is an ideal of J (see [14], p.144). We define an r-tad {p_{1} ⋯ p_{r}} for p_{1}, …, p_{n} ∈ FSJ (X) as

{p_{1} ⋯ p_{r}} = p_{1} ⋯ p_{r} + p_{r} ⋯ p_{1}.

In particular, {p_{1}p_{2}p_{3}p_{4}} is called a tetrad. A formal ideal H ◁ FSJ (X) is called hermitian if it is closed under tetrads, i.e.,

{HHHH} ⊂ H.

An ideal I ◁ FSJ (X) is called a linearization-invariant T-ideal if I contains all the linearizations of any p ∈ I and T (I) ⊂ I for any algebra endomorphism T of FSJ (X). Let Q_{48} be the linearization-invariant T -ideal of FSJ (X) generated by q_{48}. Then it is known that Q_{48} is an hermitian ideal (see [14] p.198).

Now, let J be any strongly prime special Jordan algebra with q_{48}(J) ≠ {0}. Then the evaluation Q_{48}(J) is a nonzero ideal of J. Since J is special, there exists an associative algebra A with involution ∗ such that J ⊂ H(A, ∗) = {a ∈ A | a^{∗} = a}. Let P be the associative subalgebra of A generated by Q_{48}(J). By the Special Hermitian Structure Theorem [14] p.146, one has H(P, ∗) = Q_{48}(J) and P is ∗- prime.

Lemma 4.1. Let J be a Jordan torus over F of Hermitian type. Then J = H(P, ∗) for some ∗-prime associative algebra P and P is generated by J.

Proof. By the observation above, we already know

H(P, ∗) = Q_{48}(J) ◁ J

for some ∗-prime associative algebra P and P is generated by Q_{48}(J). Thus we only need to show Q_{48}(J) = J. Let B be a basis of J over F such that B consists of homogeneous elements in J. Recall that Q_{48} contains q_{48} and all the linearizations of q_{48}. If the evaluations of q_{48} and all the linearizations of q_{48} on B vanish, then we have q_{48}(J) = {0}, which contradicts the fact that J is of Hermitian type. Hence there exist elements b_{1}, …, b_{m} ∈ B and q′_{48} ∈ Q_{48} where q′_{48} = q_{48} or some linearization of q_{48} such that q′_{48}(b_{1}, …, b_{m}) ≠ 0. Since q′_{48} is homogeneous in each variable, q′_{48}(b_{1}, …, b_{m}) is a nonzero homogeneous element in J, and hence it is invertible in J. Thus the ideal Q_{48}(J) contains an invertible element, and so we get Q_{48}(J) = J. □

Definition 4.2. A Jordan torus J is called an Hermitian torus if J = H(P, ∗) for some ∗-prime associative algebra P and P is generated by J.

We note {Hermitian tori} ⊃ {Jordan tori of Hermitian type}, but the other inclusion does not hold, e.g. the algebra of Laurent polynomials F_{1} = F [t_{1}^{±1}, …, t_{n}^{±1}] is not of Hermitian type since q_{48}(F_{1}) = {0}. However, F_{1} = H(F_{1}, id) is clearly a Hermitian torus.

Example 4.3. (1) The Jordan torus F_{q}^{+} (see Example 3.2) is a Hermitian torus. In fact, let P = F_{q} ⊕ F_{q}^{op} be the associative algebra, where F_{q}^{op} is the opposite algebra of F_{q}, and let ∗ be the exchange involution of P. Then F_{q}^{+} ≅ H(P, ∗) and one can easily check that P satisfies the conditions unless q = 1.

(2) Let ε = (ε_{ij}) be a quantum matrix such that ε_{ij} = 1 or −1. We call such an ε elementary. For the quantum torus F_{ε} = F_{ε}[t_{1}^{±1}, …, t_{n}^{±1}], there exists a unique involution ∗ on F_{ε} such that t_{i}^{∗} = t_{i} for all i. Thus the symmetric elements J := H(F_{ε}, ∗) form a Jordan algebra. Since ∗ is graded relative to a ⟨σ_{1}, …, σ_{n}⟩-grading of F_{ε}, J = ⊕_{α∈Λ} (F t_{α} ∩J) is a Λ-graded algebra and dim_{F} (F t_{α} ∩J) ≤ 1. In general, the inverse of a symmetric element is also symmetric. Since t_{1}, …, t_{n} ∈ J, J generates F_{ε} and supp J generates Λ. Thus J = H(F_{ε}, ∗) is a Hermitian n-torus. If q ≠ 1, then (t_{i}t_{j})^{∗} = −t_{i}t_{j} for some i, j, and so we have F t_{i}t_{j} ∩ J = {0}. Hence σ_{i} + σ_{j} ∉ supp J. Therefore, supp J = Λ if and only if q = 1, i.e., J = F [t_{1}^{±1}, …, t_{n}^{±1}]. In particular, J is never of strong type unless q = 1. We call the grading of H(F_{ε}, ∗) induced from a toral grading of F_{ε} a toral grading of H(F_{ε}, ∗). (In [2] p.16, H(F_{ε}, ∗) is used to construct EALA’s of type C.)

(3) Let E be a quadratic field extension of F. Let σ_{E} be the non-trivial Galois automorphism of E over F. Let ξ = (ξ_{ij}) be a quantum matrix over E such that

(4.4)  σ_{E}(ξ_{ij})ξ_{ij} = 1 (⟺ σ_{E}(ξ_{ij}) = ξ_{ji}) for all i, j.

For the quantum torus E_{ξ} = E_{ξ}[t_{1}^{±1}, …, t_{n}^{±1}] over E, there exists a unique σ_{E}- semilinear involution σ on E_{ξ} over F such that σ(t_{i}) = t_{i} for all i. Thus the symmetric elements H(E_{ξ}, σ) form a Jordan algebra over F, and the Λ-grading induced from a toral grading of E_{ξ} makes H(E_{ξ}, σ) into a Λ-graded algebra. One can easily check that H(E_{ξ}, σ) is a Hermitian torus over F with supp H(E_{ξ}, σ) = Λ. We call the grading of H(E_{ξ}, σ) induced from a toral grading of E_{ξ} a toral grading of H(E_{ξ}, σ). Also, we will identify E ⊗_{F} H(E_{ξ}, σ) with E_{ξ}^{+} via x ⊗ t ↔ xt for x ∈ E and t ∈ H(E_{ξ}, σ).

The following lemma is known for Jordan division algebras in [9] p.8.24. This is true for Jordan domains, and the proof is the same. But for the convenience of the reader, we prove it.

Lemma 4.5. Let J be a Jordan domain satisfying J = H(P, ∗) for an associative algebra P with involution ∗ such that P is generated by J. Suppose that there exists v ∈ P such that vv^{∗} = 0 and v + v^{∗} is invertible in J. Then J ≅ B^{+} for some associative algebra B.

Proof. For any y ∈ J = H(P, ∗) we claim that v^{∗}yv = v^{∗}v = vyv^{∗} = 0. Clearly we have v^{∗}yv ∈ H(P, ∗) = J. Also, U_{v^{∗}yv}1 = (v^{∗}yv)^{2} = (v^{∗}yv) · (v^{∗}yv) = v^{∗}yvv^{∗}yv = 0 since vv^{∗} = 0. Hence v^{∗}yv = 0 since J is a Jordan domain. In particular, v^{∗}v = 0 for y = 1. By the same argument, we get vyv^{∗} = 0, and so our claim is settled.

Now, since v+v^{∗} is invertible in J, there exists z ∈ J such that (v+v^{∗})z(v+v^{∗}) = 1. By the claim, we have e + e^{∗} = 1 where e := vzv and ee^{∗} = e^{∗}e = 0. Also, we have e = e(e + e^{∗}) = e^{2} and e^{∗} = e^{∗}(e + e^{∗}) = e^{∗}^{2}. Thus e and e^{∗} are supplementary orthogonal idempotents in P. By the claim, we have e^{∗}Je = eJe^{∗} = {0}. Since J generates P, we get e^{∗}P e = eP e^{∗} = {0}. Therefore, for the associative algebra B := eP e, we have P = B ⊕ B^{∗}. Since J = H(P, ∗), we obtain an isomorphism of Jordan algebras f : J = {b + b^{∗} | b ∈ B} ˜ ⟶B^{+}. □

We write A ≅_{Λ} B if A and B are Λ-graded isomorphic. Also, we write A ≅_{Λ} F_{q}, F_{q}^{+}, H(F_{ε}, ∗) or H(E_{ξ}, σ), if A is Λ-graded isomorphic to one of them ‘for some toral grading’.

Lemma 4.6. Let J = ⊕_{α∈Λ} J_{α} be a Jordan torus over F. If J ≅ B^{+} for some associative algebra B, then B is an associative torus and J ≅_{Λ} F_{q}^{+} for some q.

Proof. Let f : J ˜ ⟶B^{+} be an isomorphism and B_{α} := f(J_{α}) for all α ∈ Λ. Then B^{+} = ⊕_{α∈Λ} B_{α} is a Jordan torus such that J ≅_{Λ} B^{+}. We show that B = ⊕_{α∈Λ} B_{α} is an associative torus. Since supp B = supp B^{+}, we have Λ = ⟨supp B^{+}⟩ = ⟨supp B⟩. Also, all nonzero elements of B_{α} are invertible in B^{+}, and so are they in B. Since dim_{F} B_{α} = dim_{F} J_{α} ≤ 1, we only need to show that B_{α}B_{β} ⊂ B_{α+β} for all α, β ∈ Λ. Note that we have B_{α} · B_{β} = B_{α} ∘ B_{β} ⊂ B_{α+β}. If B_{α} = (0) or B_{β} = (0), we have nothing to prove. Otherwise, for 0 ≠ x ∈ B_{α} and 0 ≠ y ∈ B_{β}, xy and yx are invertible in B and so are they in B^{+}. Hence, by 3.6(ii), xy ∈ B_{γ} and yx ∈ B_{δ} for some γ, δ ∈ Λ. If x ∘ y = xy + yx ≠ 0, then 0 ≠ x ∘ y ∈ B_{α+β} ∩ (B_{γ} + B_{δ}), which forces α + β = γ = δ. So we get xy ∈ B_{α+β}. If x ∘ y = xy + yx = 0, i.e., yx = −xy, then we have [x^{2}, y^{2}] = 0, and so x^{2} · y^{2} = x^{2}y^{2}. Thus we get

0 ≠ (xy)^{2} = (xy) · (xy) = xyxy = −x^{2}y^{2} = −x^{2} · y^{2} ∈ B_{2γ} ∩ B_{2α+2β}.

Hence γ = α + β and we obtain xy ∈ B_{α+β}. Therefore, B = ⊕_{α∈Λ} B_{α} is an associative torus, and so B ≅_{Λ} F_{q} for some q and we get J ≅_{Λ} F_{q}^{+}. □

Since Jordan tori are Jordan domains, one gets (a) of the following by 4.5 and 4.6:

Proposition 4.7. Let J = ⊕_{α∈Λ} J_{α} be a Jordan torus over F satisfying J = H(P, ∗) for an associative algebra P with involution ∗ such that P is generated by J.

(a) Suppose that there exists v ∈ P such that vv^{∗} = 0 and v + v^{∗} is invertible in J. Then J ≅_{Λ} F_{q}^{+} for some q.

(b) Suppose that there exist an invertible element u ∈ P so that u^{∗} = −u and 0 ≠ y ∈ J_{γ} for some γ ∈ Λ such that the following three conditions hold:

(i) u^{2} ∈ J_{2γ}, (ii) uy^{−1}u ∈ J_{γ}, (iii) [u, y] ∈ J_{2γ}.

Then J ≅_{Λ} F_{q}^{+} or E ⊗_{F} J ≅_{Λ} E_{q}^{+} for some q where E is a quadratic field extension of F.

Proof. We only need to show (b). By dim_{F} J_{α} ≤ 1 for all α ∈ Λ, there exist a, b, c ∈ F, a, b ≠ 0, such that

(1)  u^{2} = ay^{2},
(2)  buy^{−1}u = y (⟺ yu^{−1} = buy^{−1}),
(3)  [u, y] = uy − yu = cu^{2}.

By (3), we have uyu^{−1} − y = cu. By (2), we have bu^{2}y^{−1} − y = cu. By (1), we have aby^{2}y^{−1} − y = cu. Hence we get cu = aby − y ∈ J. Since u ∉ J, we obtain c = 0 and hence

(4)  uy = yu.

Let v := u+\sqrt{a} y if \sqrt{a} ∈ F. Otherwise, let E := F (\sqrt{a}), J_{E} := E⊗_{F} J, P_{E} := E⊗_{F} P,

∗ := id ⊗∗ and v := 1 ⊗ u + \sqrt{a} ⊗ y = u + \sqrt{a} y. Then since u^{∗} = −u, we have

vv^{∗} = \{ \begin{matrix}(u + \sqrt{a} y)(−u + \sqrt{a} y) = −u^{2} + ay^{2} = 0 \\ (1 ⊗ u + \sqrt{a} ⊗ y)(−1 ⊗ u + \sqrt{a} ⊗ y) = 1 ⊗ (−u^{2} + ay^{2}) = 0, \end{matrix}

by (4) and (1). Also, v + v^{∗} = 2\sqrt{a} y or 2\sqrt{a} ⊗ y is invertible in J or in J_{E}. Thus, by (a), we get J ≅_{Λ} F_{q}^{+} if \sqrt{a} ∈ F. If \sqrt{a} ∉ F, then we can apply (a) for the Jordan torus J_{E} = H(P_{E}, ∗) over E, and obtain J_{E} ≅_{Λ} E_{q}^{+}. □

For the next proposition, we need the following fact:

4.8. If A and B are associative algebras, B has no zero-divisors and f : A^{+} ⟶ B^{+} is a homomorphism of Jordan algebras, then f : A ⟶ B is either a homomorphism or anti-homomorphism of associative algebras ([9] Theorem 1.1.7, p.1.4). □

Proposition 4.9. Let E ⊃ F be fields with [E : F] = 2, and σ_{E} is the nontrivial Galois automorphism of E over F. Let q ∈ M_{n}(E) be a quantum matrix over E.

(1) Suppose that τ is a σ_{E}-semilinear involution of E_{q}^{+} = E_{q}[t_{1}^{±1}, …, t_{n}^{±1}]^{+} over F such that τ(t_{i}) = t_{i} for all i. Then:

(i) q ∈ M_{n}(F) and τ is an automorphism of E_{q} over F or

(ii) q = ξ and τ is an involution of E_{ξ} over F, where ξ satisfies (4.4). In particular, τ = σ where σ is defined in 4.3(3).

(2) Let J be a Jordan torus over F. Suppose that J_{E} = E ⊗_{F} J ≅_{Λ} E_{q}^{+}. Then:

(i) q ∈ M_{n}(F) and J ≅_{Λ} F_{q}^{+} or

(ii) J ≅_{Λ} H(E_{ξ}, σ), which is a Hermitian torus defined in 4.3(3). In this case we can identify E ⊗_{F} H(E_{ξ}, σ) = E_{ξ}^{+} so that σ_{E} ⊗ id = σ.

Proof. (1): Since E_{q} has no zero-divisors, τ is an order 2 automorphism or an involution of the associative F -algebra E_{q} (see 4.8). If τ is an automorphism, then

t_{j}t_{i} − q_{ij}t_{i}t_{j} = 0 = τ(t_{j}t_{i} − q_{ij}t_{i}t_{j}) = t_{j}t_{i} − σ_{E}(q_{ij})t_{i}t_{j},

which forces σ_{E}(q_{ij}) = q_{ij}, and so q_{ij} ∈ F for all i, j, i.e., q ∈ M_{n}(F).

If τ is an involution, then

0 = τ(t_{j}t_{i} − q_{ij}t_{i}t_{j}) = t_{i}t_{j} − σ_{E}(q_{ij})t_{j}t_{i} = t_{i}t_{j} − σ_{E}(q_{ij})q_{ij}t_{i}t_{j}.

Hence we get σ_{E}(q_{ij})q_{ij} = 1 for all i, j and obtain q = ξ. In particular, τ = σ since σ is a unique σ_{E}-semilinear involution of E_{ξ} such that σ(t_{i}) = t_{i} for all i.

(2): Let τ := σ_{E} ⊗ id, which is a σ_{E}-semilinear involution of J_{E} = E ⊗_{F} J. Identifying J with 1 ⊗ J, we have J = H(J_{E}, τ), the set of fixed points by τ. Also, we identify J_{E} with E_{q}^{+}, and so J = H(E_{q}^{+}, τ). Since supp J = supp J_{E} = supp E_{q}^{+} = Λ, one can choose t_{1}, …, t_{n} ∈ E_{q} such that E_{q} = E_{q}[t_{1}^{±1}, …, t_{n}^{±1}] and τ(t_{i}) = t_{i} for all i. Thus one can apply (1). For the case (i), one gets H(E_{q}, τ) = F_{q}[t_{1}^{±1}, …, t_{n}^{±1}], and so J = F_{q}^{+}. For the case (ii), one obtains q = ξ, τ = σ and J = H(E_{ξ}, σ). □

We note the following fact about semilattices:

4.10. Let S be a semilattice in Λ. Then there exists a basis {σ_{1}, ⋯, σ_{n}} of Λ such that each σ_{i} ∈ S ([1] p.24 Proposition 1.11). □

The reader is reminded that any Jordan division algebra of Hermitian type is isomorphic to A^{+} or H(B, ∗) for some associative division algebra A or some associative division algebra B with involution ∗. We are now ready to prove an analogous result for Hermitian tori.

Theorem 4.11. Any Hermitian torus over F is graded isomorphic to one of the three tori F_{q}^{+}, H(F_{ε}, ∗) or H(E_{ξ}, σ) for some toral grading, as described in 4.3, and conversely, these three tori are all Hermitian tori.

Proof. We only need to show the first statement. Let J = ⊕_{α∈Λ} J_{α} be a Hermitian torus over F, i.e., J = H(P, ∗) for some ∗-prime associative algebra P which is generated by J. Let ⟨σ_{1}, …, σ_{n}⟩ be a basis of Λ such that each σ_{i} ∈ supp J (see 4.10), and let 0 ≠ x_{i} ∈ J_{σ_{i}}, i = 1, …, n. We first consider the case where the following two conditions hold:

(A) for all 1 ≤ i, j ≤ n, [x_{i}, x_{j}] = 0 or x_{i} ∘ x_{j} = 0, i.e., x_{j}x_{i} = ±x_{i}x_{j} for all

1 ≤ i, j ≤ n, (B) J is generated by r-tads {x_{i_{1}}^{ε_{1}} ⋯ x_{i_{r}}^{ε_{r}}} where r > 0, i_{1}, …, i_{r} ∈ {1, …, n} and

ε_{k} = ±1. We will show J ≅_{Λ} H(F_{ε}, ∗) in this case. Since J generates P and every r-tad is generated by 1-tads 2x_{1}^{±1}, …, 2x_{n}^{±1} as an associative algebra, P is generated by x_{1}^{±1}, …, x_{n}^{±1}. Thus, by (A), there exist an elementary quantum matrix ε and an epimorphism φ from F_{ε} = F_{ε}[t_{1}^{±}, …, t_{n}^{±}] onto P such that φ(t_{i}) = x_{i} for i = 1, …, n. We give F_{ε} the ⟨σ_{1}, …, σ_{n}⟩-grading, and show injectivity of φ. Suppose that φ(t) = 0 for t = ∑_{α∈Λ} a_{α}t_{α} where a_{α} ∈ F, t_{α} = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}} and α = α_{1}σ_{1} + ⋯ + α_{n}σ_{n} ∈ Λ. So we have

(1)  ∑_{α∈Λ} a_{α}x_{α} = 0,

where x_{α} := x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}. Note that all x_{α} are invertible in P since x_{i} is invertible in J and in P. We need to show that all a_{α} = 0. Put

M := {α ∈ Λ | x_{α} ∈ J} and N := {β ∈ Λ | x_{β} ∉ J}.

Then we have ∑_{α∈M} a_{α}x_{α} = 0 and ∑_{β∈N} a_{β}x_{β} = 0 since x_{α} ∈ J are symmetric and x_{β} ∉ J are skew relative to ∗ (note ch. F ≠ 2).

Claim 1. Assuming only (A), we have x_{α} ∈ J ⟹ x_{α} ∈ J_{α}.

Proof. By (A), the subalgebra of A generated by {x_{i}^{2}}_{i=1}^{n} is commutative, and so the Jordan product and the associative product coincide in the subalgebra. Therefore,

x_{α}^{2} = ±x_{1}^{2α_{1}} ⋯ x_{n}^{2α_{n}} = ±(· · (x_{1}^{2α_{1}} · x_{2}^{2α_{2}}) ⋯ x_{n}^{2α_{n}}) ∈ J_{2α_{1}σ_{1}+···+2α_{n}σ_{n}} = J_{2α}.

Hence, by 3.7, we get x_{α} ∈ J_{α}. □

By Claim 1, we obtain a_{α} = 0 for all α ∈ M. If N = ∅, we are done. Otherwise we pick any β_{0} ∈ N. Multiply (1) by x_{β_{0}}. Then, by (A), we have

∑_{β∈N} ±a_{β}x_{β+β_{0}} = 0.

Applying the same argument for this equation instead of (1), we have

∑_{β∈M_{1}} ±a_{β}x_{β+β_{0}} = 0 and ∑_{γ∈N_{1}} ±a_{γ}x_{γ+β_{0}} = 0,

where M_{1} := {β ∈ N | x_{β+β_{0}} ∈ J} and N_{1} := {γ ∈ N | x_{γ+β_{0}} ∉ J}. By Claim 1, we get a_{β}x_{β+β_{0}} ∈ J_{β+β_{0}}, and hence a_{β} = 0 for all β ∈ M_{1}. Since N = M_{1} ⊔ N_{1} (disjoint union) and β_{0} ∈ M_{1} (because x_{2β_{0}} ∈ J), we have N_{1} ⊊ N. If N_{1} = ∅, we are done. Otherwise, repeating this method for the finite set N_{1}, we get some r > 1 such that N_{r} = ∅ and a_{β} = 0 for all β ∈ M_{2} ⊔ ⋯ ⊔ M_{r} = N_{1}. Consequently, we obtain a_{α} = 0 for all α ∈ Λ. Thus t = 0 and φ is injective.

We have shown that φ is an isomorphism. Further, P is graded with P_{σ_{i}} = J_{σ_{i}} and φ is a graded isomorphism. Also, through this isomorphism, we get an involution ∗ of F_{ε} such that t_{i}^{∗} = t_{i} for i = 1, …, n. Therefore, we obtain J ≅_{Λ} H(F_{ε}, ∗).

We consider the second case: the negation of (A), i.e.,

there exist some i, j such that u := [x_{i}, x_{j}] ≠ 0 and x_{i} ∘ x_{j} ≠ 0.

We divide the case into two subcases:

(I) u^{2} = 0 and (II) u^{2} ≠ 0.

Note that u^{∗} = −u.

(I): We have uu^{∗} = −u^{2} = 0. We need the following claim which can be proven in the same manner as in the classification of Jordan division algebras (see [9] p.8.25).

Claim 2. There exists y ∈ J such that for v = yu, v + v^{∗} ≠ 0.

Proof. Otherwise, for all y ∈ J, we have v +v^{∗} = 0 for v = yu, i.e., yu = −u^{∗}y^{∗} = uy. So for all w ∈ P, we have (uy)(uw) = u^{2}yw = 0, and hence we have (uJ)(uP) = {0}. Since P is generated by J, we get (uP)^{2} = {0}. Then we have (P uP)^{2} = P uP P uP = P uP uP = P (uP)^{2} = {0}. Moreover, (P uP)^{∗} = P u^{∗}P = P uP, and so P uP is a nonzero ∗-ideal. This contradicts the fact that P is ∗-prime (∗-semiprime is enough!). □

Let y ∈ J be such an element as in Claim 2 so that

v + v^{∗} = yu − uy = [y, u] ≠ 0.

Decompose y into nonzero distinct homogeneous elements, namely, y = ∑_{α} y_{α}. Let v_{α} := y_{α}u for all y_{α}. Suppose that v_{α} + v_{α}^{∗} = 0 for all v_{α}. Then we have [y_{α}, u] = y_{α}u − uy_{α} = v_{α} + v_{α}^{∗} = 0 for all y_{α}. Therefore,

v + v^{∗} = [y, u] = [ ∑_{α} y_{α}, u] = ∑_{α} [y_{α}, u] = 0,

which contradicts our choice of y. Hence there exists some v_{α} such that v_{α} + v_{α}^{∗} ≠ 0. By 2.7, we have

0 ≠ v_{α} + v_{α}^{∗} = [y_{α}, u] = [y_{α}, [x_{i}, x_{j}]] = (x_{i}, y_{α}, x_{j})^{∘} ∈ J_{σ_{i}+α+σ_{j}}.

Hence v_{α} + v_{α}^{∗} is invertible in J. Also, we have v_{α}v_{α}^{∗} = y_{α}uu^{∗}y_{α}^{∗} = −y_{α}u^{2}y_{α} = 0. Therefore, by 4.7(a), we get J ≅_{Λ} F_{q}^{+} for some q.

(II): Let 0 ≠ y := x_{i} ∘ x_{j} ∈ J_{γ} where γ := σ_{i} + σ_{j}. We show that these u and y satisfy the three conditions in 4.7(b). By 2.8, we have

0 ≠ u^{2} = [x_{i}, x_{j}]^{2} = x_{i} ∘ U_{x_{j}}x_{i} − U_{x_{i}}x_{j}^{2} − U_{x_{j}}x_{i}^{2} ∈ J_{2γ}.

Hence u^{2} is invertible in J and hence in P. Thus u is invertible in P and u^{2} ∈ J_{2γ}. By 2.9, we have

uy^{−1}u = U_{u}y^{−1} = U_{[x_{i},x_{j}]}y^{−1} = (U_{x_{i}∘x_{j}} − 2U_{x_{i}}U_{x_{j}} − 2U_{x_{j}}U_{x_{i}})y^{−1} ∈ J_{γ} since y^{−1} ∈ J_{−γ}. By 2.7, we have

[u, y] = [[x_{i}, x_{j}], x_{i} ∘ x_{j}] = −(x_{i}, x_{i} ∘ x_{j}, x_{j})^{∘} ∈ J_{2γ}.

Thus u and y satisfy the conditions in 4.7(b), and we get J ≅_{Λ} F_{q}^{+} or J_{E} ≅_{Λ} E_{q}^{+}. Then, by 4.9(2), we obtain J ≅_{Λ} F_{q}^{+} or H(E_{ξ}, σ).

We consider the final case: (A) with the negation of (B), i,e.,

assuming the relation (A), J is not generated by r-tads {x_{i_{1}}^{ε_{1}} ⋯ x_{i_{r}}^{ε_{r}}}. By our assumption, there exist γ ∈ Λ and 0 ≠ y ∈ J_{γ} such that y is not generated by r-tads {x_{i_{1}}^{ε_{1}} ⋯ x_{i_{r}}^{ε_{r}}}. Let γ = γ_{1}σ_{1} + ⋯ + γ_{n}σ_{n}.

Claim 3. u := x_{1}^{γ_{1}} ⋯ x_{n}^{γ_{n}} ∉ J.

Proof. Otherwise we have 2u = {x_{1}^{γ_{1}} ⋯ x_{n}^{γ_{n}}}, and by Claim 1, u ∈ J_{γ}. So we get y = au for some a ∈ F by dim_{F} J_{γ} ≤ 1, i.e., y = \frac{1}{2}a\{x_{1}^{γ_{1}} ⋯ x_{n}^{γ_{n}}\}, which contradicts our setting of y. □

Now, we show that these u and y ∈ J_{γ} satisfy the conditions in 4.7(b). Observe first that u^{∗} = ±u by (A) and hence u^{∗} = −u by Claim 3. Next, by definition, u is clearly invertible, and by (A), we have u^{2} = ±x_{1}^{2γ_{1}} ⋯ x_{n}^{2γ_{n}} ∈ J. Hence u^{2} ∈ J_{2γ} by Claim 1. Secondly we have

U_{u}y^{−1} = uy^{−1}u = x_{1}^{γ_{1}} ⋯ x_{n}^{γ_{n}}y^{−1}x_{1}^{γ_{1}} ⋯ x_{n}^{γ_{n}} = ±x_{1}^{γ_{1}} ⋯ x_{n}^{γ_{n}}y^{−1}x_{n}^{γ_{n}} ⋯ x_{1}^{γ_{1}}
= ±U_{x_{1}^{γ_{1}}} ⋯ U_{x_{n}^{γn}} y^{−1} ∈ J_{γ}.

In particular, we get a formula for u:

(2)  U_{u} = ±U_{x_{1}^{γ_{1}}} ⋯ U_{x_{n}^{γn}} .

Thirdly, since u^{∗} = −u and [u, y]^{∗} = [y^{∗}, u^{∗}] = −[y, u] = [u, y], we have [u, y] ∈ J. Also, we have by 2.8 and (2),

[u, y]^{2} = y ∘ U_{u}y − U_{y}u^{2} − U_{u}y^{2} ∈ J_{4γ}.

Hence, by 3.7, we get [u, y] ∈ J_{2γ}. Thus, by 4.7(b) and 4.9(2), we get J ≅_{Λ} F_{q}^{+} or H(E_{ξ}, σ). □

§5 Clifford type

Definition 5.1. A Jordan torus of Clifford type, i.e., the central closure is a Jordan algebra of a symmetric bilinear form, is called a Clifford torus.

Example 5.2. Let Λ = Λ_{n} be a free abelian group of rank n ≥ 2. Let 2 ≤ m ≤ n and let Λ_{m} and Λ_{n−m} be subgroups of Λ of rank m and n−m, respectively, such that Λ = Λ_{m} ⊕ Λ_{n−m}. Let S^{(m)} be a semilattice in Λ_{m} and {σ_{1}, …, σ_{m}} a basis of Λ_{m}. Extend {σ_{1}, …, σ_{m}} to a basis {σ_{1}, …, σ_{m}, σ_{m+1}, …, σ_{n}} of Λ. Let

Γ := 2Λ_{m} ⊕ Λ_{n−m} and Z := F [z_{1}^{±1}, …, z_{n}^{±1}] with a toral Γ-grading, i.e.,

Z = ⊕_{γ∈Γ} F z_{γ} where

z_{γ} = z_{1}^{γ_{1}} ⋯ z_{n}^{γ_{n}} for γ = 2γ_{1}σ_{1} + ⋯ + 2γ_{m}σ_{m} + γ_{m+1}σ_{m+1} + ⋯ + γ_{n}σ_{n}.

We fix

a representative set I of (S^{(m)}/2Λ_{m}) \ \{\overline{0}\}.

Let V be a free Z-module with basis {t_{ϵ}}_{ϵ∈I}. Define a Z-bilinear form f : V ×V ⟶ Z by

(∗)  f(t_{ϵ}, t_{η}) = \{ \begin{matrix}a_{ϵ}z_{2ϵ} & if ϵ = η \\ 0 & otherwise \end{matrix}

for all ϵ, η ∈ I, where 0 ≠ a_{ϵ} ∈ F. Let

J = J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}) := Z ⊕ V

be the Jordan algebra over Z of f. Then

{z_{γ} | γ ∈ Γ} ∪ {z_{γ}t_{ϵ} | γ ∈ Γ, ϵ ∈ I}

is an F -basis of J. For α ∈ S^{(m)} ⊕ Λ_{n−m}, there exist unique α′ ∈ Γ and ϵ ∈ I ∪ {0} such that α = α′ + ϵ. Put t_{0} := 1 and

t_{α} = \{ \begin{matrix}z_{α′}t_{ϵ} & if α ∈ S^{(m)} ⊕ Λ_{n−m} \\ 0 & otherwise. \end{matrix}

Then we get J = ⊕_{α∈Λ} F t_{α} as a graded F -vector space. By (∗), we have, for α = α′ + ϵ, β = β′ + η ∈ S^{(m)} ⊕ Λ_{n−m} where α′, β′ ∈ Γ, ϵ, η ∈ I ∪ {0},

(∗∗)  t_{α}t_{β} = (z_{α′}t_{ϵ})(z_{β′}t_{η}) = \{ \begin{matrix}a_{ϵ}z_{α′+β′}z_{2ϵ} = a_{ϵ}t_{α+β} & if ϵ = η ≠ 0 \\ z_{α′+β′}t_{ϵ} = t_{α+β} & if η = 0 \\ z_{α′+β′}t_{η} = t_{α+β} & if ϵ = 0 \\ 0 & otherwise, \end{matrix}

and so we obtain t_{α}t_{β} ⊂ F t_{α+β} for all α, β ∈ Λ. For α = α′+ϵ ∈ S^{(m)}⊕Λ_{n−m}, since t_{α}^{2} = a_{ϵ}z_{2α′}z_{2ϵ} is invertible, t_{α} is invertible. Since supp J = S^{(m)} ⊕ Λ_{n−m}, supp J generates Λ and hence J = ⊕_{α∈Λ} F t_{α} is a Jordan torus over F. One can check that the centre Z(J) of J is Z and the central closure is a Jordan algebra of the extended bilinear form of f. Hence J is a Clifford torus.

We call J = J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}) the Clifford torus determined by S^{(m)} of type {a_{ϵ}}_{ϵ∈I}. The Λ-grading of J determined by a basis ⟨σ_{1}, …, σ_{n}⟩ of Λ is called a toral grading or a ⟨σ_{1}, …, σ_{n}⟩-grading of J. Also, when a_{ϵ} = 1 for all ϵ ∈ I, we call the J = J_{S^{(m)}} the standard Clifford torus determined by S^{(m)}.

Remark 5.3. (1) One can easily check that

J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}) ≅_{Λ} J′ ⊗_{F} F [z_{m+1}^{±1}, …, z_{n}^{±1}]

where J′ is the Clifford torus as constructed in 5.2 for n = m. When m = n, the standard Clifford torus J_{S^{(n)}} appeared in [1] as the first example of an EALA of type A_{1} graded by an arbitrary semilattice.

(2) If any element of F has a square root in F, e.g. F is an algebraically closed field, then one can make a_{ϵ} = 1 for all ϵ ∈ I by switching t_{ϵ} to (\sqrt{a_{ϵ}})^{−1}t_{ϵ}. Thus, for such a base field F, a Clifford torus J = J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}) is always graded isomorphic to the standard Clifford torus J_{S^{(m)}}.

(3) A Clifford torus J = J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}) is, by (∗∗), never of strong type, even if we take S^{(m)} = Λ_{m}.

We now start the classification. Let J = ⊕_{α∈Λ} J_{α} be a Clifford torus over F, i.e., the central closure \overline{J} is a Jordan algebra over \overline{Z} of a symmetric bilinear form where Z = Z(J) is the centre of J and \overline{Z} is the field of fractions of Z. Thus \overline{J} has degree ≤ 2 over \overline{Z}, i.e., there exist a \overline{Z}-linear form tr : \overline{J} ⟶ \overline{Z} and a \overline{Z}-quadratic map n : \overline{J} ⟶ \overline{Z} with n(1) = 1 such that for all x ∈ \overline{J},

x^{2} − tr(x)x + n(x)1 = 0.

If dim\overline{Z} \overline{J} = 1, then J = Z since J embeds into \overline{J}. Hence J is a commutative associative torus, and so J ≅_{Λ} F [t_{1}^{±1}, …, t_{n}^{±1}].

Claim 1. Let J = ⊕_{α∈Λ} J_{α} be a Clifford torus such that dim\overline{Z} \overline{J} ≠ 1. Let Γ be the central grading group of J. Then, for any α ∈ Λ \ Γ, we have tr(J_{α}) = {0}, and there exists a basis {σ_{1}, …, σ_{n}} of Λ such that

Γ = 2ℤσ_{1} + ⋯ + 2ℤσ_{m} + ℤσ_{m+1} + ⋯ + ℤσ_{n} for some 2 ≤ m ≤ n.

Proof. Recall that \overline{J} = ⊕_{\overline{α}∈Λ/Γ} \overline{J}\overline{α} is a Λ/Γ-graded algebra over \overline{Z} (see 3.9(iii)). Note that dim\overline{Z} \overline{J} ≠ 1 implies Λ/Γ ≠ \{\overline{0}\}, and so supp \overline{J} ≠ \{\overline{0}\}. For any α ∈ supp J \ Γ, let 0 ≠ x ∈ J_{α}. Then x^{2} + n(x)1 = tr(x)x ∈ \overline{J}\overline{α}. If tr(x) ≠ 0, then 2\overline{α} = \overline{α} since x^{2} ≠ 0. Hence \overline{α} = \overline{0}, which contradicts our choice of α. So we get tr(x) = 0. Then x^{2} = −n(x)1 ∈ \overline{Z}1. Since 0 ≠ x^{2} ∈ J_{2α} ∩ \overline{Z}1 ⊂ \overline{J}_{2\overline{α}} ∩ \overline{J}\overline{0}, we get 2\overline{α} = \overline{0}.

Therefore, 2Λ ⊂ Γ, and the exponent of Λ/Γ is 2, i.e., Λ/Γ ≅ ℤ_{2}^{m} for some 1 ≤ m ≤ n. By 3.6(iv), m = 1 cannot happen. Thus the statement is clear by the Fundamental Theorem of finitely generated abelian groups. □

Let W := \{x ∈ \overline{J} | tr(x) = 0\}. Then \overline{J} = \overline{Z}1 ⊕ W is a Jordan algebra over \overline{Z} of the symmetric bilinear form

h := −\frac{1}{2}n(·, ·) |_{W ×W} .

Recall that J = ⊕_{\overline{α}∈Λ/Γ} J\overline{α} is a Λ/Γ-graded algebra over Z in 3.9(ii). By Claim 1, we have tr(J\overline{α}) = \{0\} for \overline{α} ≠ \overline{0}, and so

V := ⊕_{\overline{α}≠\overline{0}} J\overline{α} ⊂ W

and J = Z ⊕ V as a direct sum of Z-modules. For all x, y ∈ V, we have xy = h(x, y)1 ∈ J ∩ \overline{Z}1 = J ∩ Z(\overline{J}) = Z. Therefore, J = Z ⊕ V is a Jordan algebra over Z of f := h |_{V ×V}. Let S := supp J and Λ_{m} := ℤσ_{1} + ⋯ + ℤσ_{m} where {σ_{1}, …, σ_{n}} is a basis of Λ chosen in Claim 1. Claim 2. S^{(m)} := S ∩ Λ_{m} is a semilattice in Λ. Proof. Since S is a semilattice in Λ and Λ_{m} is a subgroup of Λ, we have 0, 2α − β ∈ S^{(m)} for all α, β ∈ S^{(m)}. We need to show that S^{(m)} generates Λ_{m}. For any δ = δ_{1}σ_{1} +⋯+δ_{n}σ_{n} ∈ S, we have δ′ := δ_{m+1}σ_{m+1} +⋯+δ_{n}σ_{n} ∈ Γ. Let 0 ≠ x ∈ J_{δ} and 0 ≠ z ∈ J_{δ′}. Then 0 ≠ xz^{−1} ∈ J_{δ−δ′}, and hence δ − δ′ = δ_{1}σ_{1} + ⋯ + δ_{m}σ_{m} ∈ S^{(m)}. Since S generates Λ, we have for α ∈ Λ_{m} ⊂ Λ,

α = ∑_{δ∈U} l_{δ}δ = ∑_{δ∈U} l_{δ}(δ − δ′)

where U is a finite subset of S and l_{δ} is a positive integer. Therefore, S^{(m)} generates Λ_{m}. □

We fix a representative set I of (S^{(m)}/2Λ_{m}) \ \{\overline{0}\}, as in 5.2. For ϵ ∈ I, let 0 ≠ t_{ϵ} ∈ J_{ϵ}. Then we get V = ⊕_{\overline{α}≠\overline{0}} J\overline{α} = ⊕_{ϵ∈I} Zt_{ϵ} as direct sums of Z-modules. Since Z = ⊕_{γ∈Γ} J_{γ} is a commutative associative Γ-torus, Z can be identified with F [z_{1}^{±1}, …, z_{n}^{±1}] with the toral Γ-grading as in 5.2. Moreover, for ϵ ≠ η ∈ I, we have ϵ + η ∉ Γ, and so

t_{ϵ}t_{η} = f(t_{ϵ}, t_{η}) ∈ J\overline{ϵ+η} ∩ J\overline{0} = (0).

Also, we have 0 ≠ t_{ϵ}^{2} = f(t_{ϵ}, t_{ϵ}) ∈ J_{2ϵ} = F z_{2ϵ}. Thus there exists 0 ≠ a_{ϵ} ∈ F such that f(t_{ϵ}, t_{ϵ}) = a_{ϵ}z_{2ϵ}. Hence the bilinear form f coincides with f in 5.2. Consequently, we have shown the following:

Theorem 5.5. Any J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}) as defined in Example 5.2 is a Clifford n-torus and conversely any Clifford n-torus is graded isomorphic to J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}), or to the algebra of Laurent polynomials in n variables, for some toral grading.

Also, by Remark 5.3(2) we have:

Corollary 5.6. Suppose that any element of the base field F has a square root. Then a Clifford torus is graded isomorphic to a standard Clifford n-torus J_{S^{(m)}} as defined in 5.2, or to the algebra of Laurent polynomials in n variables, for some toral grading. □

§6 Albert type

We classify Jordan tori of Albert type, i.e., Jordan tori whose central closure is an Albert algebra. An Albert algebra is defined as either a first or a second Tits construction, which are both 27-dimensional central simple exceptional Jordan algebras of degree 3. We recall the first Tits construction but not the second one since second Tits constructions do not occur in the class of Jordan tori.

Let A be a central simple associative algebra over F of (generic) degree 3 with generic trace tr. For a, b ∈ A, let

a · b = \frac{1}{2}(ab + ba),
a × b = a · b − \frac{1}{2} tr(a)b − \frac{1}{2} tr(b)a + \frac{1}{2}( tr(a) tr(b) − tr(a · b))1,
\overline{a} = a × 1 = \frac{1}{2}( tr(a)1 − a).

Note that

\overline{1} = 1 and a(a × a) = (a × a)a = n(a)

where n is the generic norm on A.

Let 0 ≠ μ ∈ F. A first Tits construction (A, μ) over F obtained from A and the structure constant μ is the direct sum A ⊕ A ⊕ A as F -spaces with the following F -bilinear multiplication:

For (a_{0}, a_{1}, a_{2}), (b_{0}, b_{1}, b_{2}) ∈ (A, μ) = A ⊕ A ⊕ A,

(a_{0}, a_{1}, a_{2})(b_{0}, b_{1}, b_{2}) = (a_{0} · b_{0} + \overline{a_{1}b_{2}} + \overline{b_{1}a_{2}},

\overline{a}_{0}b_{1} + \overline{b}_{0}a_{1} + μ^{−1}a_{2} × b_{2}, b_{2}\overline{a}_{0} + a_{2}\overline{b}_{0} + μa_{1} × b_{1}) (6.1)

We will always identify A with (A, 0, 0) as F -spaces.

Remark 6.2. Let x := (0, 1, 0) and y := (0, 0, 1). Since a = tr(a)1 + 2\overline{a} = \overline{tr(a)1 + 2a}, we have A · x = (0, A, 0) and A · x^{2} = (0, 0, μA) = (0, 0, A). Thus (A, μ) is generated by A and x. Also, since x = μy^{2}, (A, μ) is generated by A and y.

The following lemma is well-known ([8] p.422, Exercise 1).

6.3. Let (A, μ) be a first Tits construction. Let a ∈ A be invertible and x = (0, a, 0), y = (0, 0, a) ∈ (A, μ). Then:

(i) 0 ≠ x^{3} ∈ F 1, and there exists an isomorphism Φ from (A, μ) onto (A, x^{3}) over F = F 1 (identify) such that Φ |_{A}= id and Φ(x) = (0, 1, 0),

(ii) 0 ≠ y^{−3} ∈ F 1, and there exists an isomorphism Ψ from (A, μ) onto (A, y^{−3}) over F = F 1 such that Ψ |_{A}= id and Ψ(y) = (0, 0, 1). □

The theory of first Tits constructions over a ring does not seem to be much developed. As far as the author knows, the most general paper is [16]. For our purpose, we do not need this generality, but only a very special case. One might say that this is almost the classical case (i.e., Tits constructions over a field above).

Definition 6.4. We say that a prime Jordan or associative algebra P over F has central degree 3 if the central closure \overline{P} = \overline{Z} ⊗_{Z} P is finite dimensional and has (generic) degree 3 over \overline{Z}.

Lemma 6.5. Let A be a prime associative algebra over F of central degree 3, and μ ∈ Z a unit where Z = Z(A) is the centre of A. Assume that tr(A) ⊂ Z where tr is the generic trace of the central closure \overline{A} over \overline{Z}. Then, the subset (A, μ) := A ⊕ A ⊕ A of the first Tits construction (\overline{A}, μ) = \overline{A} ⊕ \overline{A} ⊕ \overline{A} is a Z-subalgebra such that \overline{(A, μ)} = (\overline{A}, μ).

Proof. By the multiplication rule of (\overline{A}, μ) (see 6.1), and our assumption tr(A) ⊂ Z, it is clear that (A, μ) is a Z-subalgebra of (\overline{A}, μ). Let Z(A, μ) be the centre of (A, μ). Then we have Z ⊂ Z(A, μ) ⊂ Z(A) = Z, and so Z(A, μ) = Z. Therefore, the central closure \overline{(A, μ)} of (A, μ) is given as \overline{Z} ⊗_{Z} (A, μ), which is a \overline{Z}-subalgebra of (\overline{A}, μ). Thus we only need to show that (\overline{A}, μ) ⊂ \overline{(A, μ)}. But this is clear because we have, for z_{i} ∈ Z and a_{i} ∈ A, i = 0, 1, 2,

(\frac{1}{z_{0}} ⊗ a_{0}, \frac{1}{z_{1}} ⊗ a_{1}, \frac{1}{z_{2}} ⊗ a_{2}) = \frac{1}{z_{0}z_{1}z_{2}} (1 ⊗ z_{1}z_{2}a_{0}, 1 ⊗ z_{0}z_{2}a_{1}, 1 ⊗ z_{0}z_{1}a_{2})

Hence we get (\overline{A}, μ) = \overline{(A, μ)}. □

We call this (A, μ) a first Tits construction over Z. This is a special type of the general first Tits construction studied in [16].

Remark 6.6. This (A, μ) is also generated as a Z-algebra by A and x = (0, 1, 0) or by A and y = (0, 0, 1) as in the classical case (see 6.2). Thus (A, μ) is characterized as the Z-subalgebra of (\overline{A}, μ) generated by A and x or by A and y.

Before giving examples of Jordan tori of central degree 3, we show general properties of them.

Proposition 6.7. Let T = ⊕_{α∈Λ} T_{α} be a division Λ-graded Jordan or associative algebra over F of central degree 3. Let tr be the generic trace of the central closure \overline{T}. Then there exists a basis \{σ_{1}, …, σ_{n}\} of Λ such that the central grading group of T is given as

Γ = 3ℤσ_{1} + ⋯ + 3ℤσ_{m} + ℤσ_{m+1} + ⋯ + ℤσ_{n} for some 2 ≤ m ≤ n,

and supp T = Λ. Moreover, for any α ∈ Λ \ Γ, we have tr(T_{α}) = {0}.

Proof. If Γ = Λ, then dim\overline{Z} \overline{T} = 1, and hence T does not have central degree 3. Therefore, Γ ≠ Λ and supp T/Γ ≠ \{0\}. Let \overline{0} ≠ \overline{β} ∈ supp T/Γ and 0 ≠ x ∈ \overline{T} \overline{β}. Since \overline{T} = ⊕_{\overline{α}∈Λ/Γ} \overline{T} \overline{α} (see 3.9(iii)), has generic degree 3, we have

x^{3} + z_{1}x^{2} + z_{2}x + z_{3}1 = 0 for some z_{1}, z_{2}, z_{3} ∈ \overline{Z} and z_{1} = − tr(x).

If \overline{2β} = \overline{0}, then \overline{3β} = \overline{β} and therefore,

x^{3} + z_{2}x = −z_{1}x^{2} − z_{3}1 ∈ \overline{T} \overline{β} ∩ \overline{T} \overline{0} = (0).

Hence we get x^{3}+z_{2}x = x(x^{2}+z_{2}1) = 0. Since \overline{T} is a Jordan or an associative domain, the subalgebra \overline{Z}[x] of \overline{T} generated by x is a commutative associative algebra domain over \overline{Z}. So x^{2}+z_{2}1 = 0 since x ≠ 0. Since x ∉ \overline{T} \overline{0}, the polynomial f(λ) = λ^{2}+z_{2} is the minimal polynomial of x over \overline{Z}. If f(λ) is reducible over \overline{Z}, say f(λ) = (λ−a)(λ−b), a, b ∈ \overline{Z}, then (x − a1)(x − b1) = 0 in \overline{Z}[x]. Hence, x = a1 or x = b1, and so x ∈ \overline{Z}1 = \overline{T} \overline{0}, i.e., \overline{β} = \overline{0}, which is a contradiction. Therefore, f(λ) is irreducible over \overline{Z}. Note that the minimal polynomial and the generic minimal polynomial of an element have the same irreducible factors (see [8] p.224). Since f(λ) is the irreducible minimal polynomial of x, the generic minimal polynomial of x has to be a power of f(λ). However, this is impossible since the degree of the generic minimal polynomial of x is 3. Therefore, \overline{2β} ≠ \overline{0}. This implies that \overline{3β} ≠ \overline{β}. Since \overline{β} ≠ \overline{0}, we have \overline{3β} ≠ \overline{2β}. Hence \{\overline{3β}, \overline{0}\} ∩ \{\overline{2β}, \overline{β}\} = ∅. So (\overline{T} \overline{3β} + \overline{T} \overline{0}) ∩ (\overline{T} \overline{2β} ⊕ \overline{T} \overline{β}) = (0). Since

x^{3} + z_{3}1 = −z_{1}x^{2} − z_{2}x ∈ (\overline{T} \overline{3β} + \overline{T} \overline{0}) ∩ (\overline{T} \overline{2β} ⊕ \overline{T} \overline{β}),

we get two equalities x^{3} + z_{3}1 = 0 and −z_{1}x^{2} − z_{2}x = 0.

By the first identity, we have 0 ≠ x^{3} = −z_{3}1 ∈ \overline{T} \overline{3β} ∩ \overline{T} \overline{0}, and hence \overline{3β} = \overline{0}. Thus 3Λ ⊂ Γ, and so the exponent of Λ/Γ is 3. Hence, by 3.9(iv), Λ/Γ ≅ ℤ_{3}^{m} for some 2 ≤ m ≤ n, and so the first statement follows from the Fundamental Theorem of finitely generated abelian groups. Also, we have 3Λ ⊂ supp T. Since supp T is a semilattice, Λ = 3Λ − 2Λ ⊂ supp T, and so supp T = Λ.

By the second identity and by the same reason above, we have −z_{1}x − z_{2}1 = 0. Then −z_{1}x = z_{2}1 ∈ \overline{T} \overline{β} ∩ \overline{T} \overline{0} = (0). Hence z_{1} = 0, i.e., tr(x) = 0. Therefore, for any α ∈ Λ \ Γ, we have tr(T_{α}) = {0}. □

We give examples of Jordan tori of central degree 3.

Example 6.8. (1) Assume that F contains a primitive 3rd root of unity ω. Let ω be an n × n (n ≥ 2) quantum matrix

ω = ω_{n} = ( \begin{matrix}1 & ω & 1 & ⋯ & 1 \\ ω^{−1} & 1 & 1 & & ⋮ \\ 1 & 1 & 1 & & ⋮ \\ ⋮ & & & ... & 1 \\ 1 & ⋯ & ⋯ & 1 & 1 \end{matrix} )

where the (1, 2)-entry is ω, the (2, 1)-entry is ω^{−1} and the other entries are all 1. Let F_{ω} = F_{ω}[u_{1}^{±1}, …, u_{n}^{±1}] be the quantum torus determined by ω and Z = Z(F_{ω}) the centre of F_{ω}. One finds that Z = F [u_{1}^{±3}, u_{2}^{±3}, u_{3}^{±1}, …, u_{n}^{±1}], the algebra of Laurent polynomials in the variables u_{1}^{3}, u_{2}^{3}, u_{3}, …, u_{n}. So for a ⟨σ_{1}, …, σ_{n}⟩-grading of F_{ω}, the central grading group of F_{ω} is 3ℤσ_{1} + 3ℤσ_{2} + ℤσ_{3} + ⋯ + ℤσ_{n}, and the central closure \overline{F}_{ω} is a (ℤ_{3} × ℤ_{3})-torus over \overline{Z}. So the dimension of \overline{F}_{ω} over \overline{Z} is 9. Since \overline{F}_{ω} is a domain, \overline{F}_{ω} is a division algebra, by Wedderburn’s Structure Theorem, and \overline{F}_{ω} has degree 3 over \overline{Z}. Thus F_{ω} has central degree 3. We claim that F_{ω}^{+} is a special Jordan torus of central degree 3. In fact, we have, by 2.5, Z(F_{ω}^{+}) = Z(F_{ω}). So the central closure \overline{F_{ω}^{+}} is a 9-dimensional central special Jordan division algebra over \overline{Z(F_{ω}^{+})} = \overline{Z(F_{ω})}. Hence, by 2.11(b), it has degree 3.

(2) Assume that n ≥ 3. Let F_{ω} = F_{ω}[u_{1}^{±1}, …, u_{n}^{±1}] as defined in (1), Z(F_{ω}) its centre and tr the generic trace of the central closure \overline{F}_{ω}. Then, by 6.7, we have tr(u_{1}^{i}u_{2}^{j}) = 0 if i ≢ 0 or j ≢ 0 (mod 3), and so

tr(F_{ω}) ⊂ Z(F_{ω}) = F [u_{1}^{±3}, u_{2}^{±3}, u_{3}^{±1}, …, u_{n}^{±1}].

Thus, by 6.5, we have the first Tits construction 𝔸_{t} = (F_{ω}, u_{3}) over Z. Namely,

𝔸_{t} = (F_{ω}, u_{3}) = F_{ω} ⊕ F_{ω} ⊕ F_{ω}.

Let ⟨σ_{1}, …, σ_{n}⟩ be a basis of Λ and

Δ := ℤσ_{1} + ℤσ_{2} + 3ℤσ_{3} + ℤσ_{4} + ⋯ + ℤσ_{n}.

We give F_{ω} a toral Δ-grading, i.e.,

F_{ω} = ⊕_{δ∈Δ} F u_{δ} where u_{δ} = u_{1}^{δ_{1}}u_{2}^{δ_{2}}u_{3}^{δ_{3}} ⋯ u_{n}^{δ_{n}}

for δ = δ_{1}σ_{1} + δ_{2}σ_{2} + 3δ_{3}σ_{3} + δ_{4}σ_{4} + ⋯ + δ_{n}σ_{n}.

For α = α_{1}σ_{1} + ⋯ + α_{n}σ_{n} ∈ Λ, we put

t_{α} := \{ \begin{matrix}(u_{α}, 0, 0) & if α_{3} ≡ 0 (mod 3) \\ (0, u_{α−σ_{3}}, 0) & if α_{3} ≡ 1 (mod 3) \\ (0, 0, u_{α+σ_{3}}) & if α_{3} ≡ 2 (mod 3). \end{matrix}

Then we obtain t_{α} ≠ 0 for all α ∈ Λ and 𝔸_{t} = ⊕_{α∈Λ}F t_{α}. Thus, 𝔸_{t} is a Λ-graded vector space over F whose homogeneous spaces are all 1-dimensional over F. We note that t_{σ_{3}} = (0, 1, 0), t_{2σ_{3}} = t_{σ_{3}}^{2} = (0, 0, u_{3}) and t_{−σ_{3}} = t_{σ_{3}}^{−1} = (0, 0, 1). By the multiplication rule of Tits first constructions (see 6.1), one can check that 𝔸_{t} is a Λ-graded algebra and the structure constants relative to the basis {t_{α}}_{α∈Λ} are

\{1, ω, ω^{2}, −\frac{1}{2}, −\frac{ω}{2} , −\frac{ω^{2}}{2} \}.

Hence 𝔸_{t} is a Jordan torus over F of strong type, which is called the Albert torus over F. We call the grading of 𝔸_{t} above a toral grading or a ⟨σ_{1}, …, σ_{n}⟩-grading. By 6.5, the central closure \overline{𝔸}_{t} of 𝔸_{t} is an Albert algebra (\overline{F}_{ω}, u_{3}) over \overline{Z}, and so the Albert torus is in fact a Jordan torus of Albert type. By 2.10, the central closure of a Jordan torus of Albert type is a division algebra since this central closure is a 27-dimensional Jordan domain. In particular, the central closure \overline{𝔸_{t}} is a division algebra.

(3) Suppose that \sqrt{−3} ∉ F. Let E = F (\sqrt{−3}) and consider the Jordan torus H(E_{ω}, σ) defined in 4.3(3). Since E⊗_{F} H(E_{ω}, σ) ≅_{Λ} E_{ω}^{+}, the central grading group of H(E_{ω}, σ) coincides with the one of E_{ω} (see 3.8), which is 3ℤσ_{1}+3ℤσ_{2}+ℤσ_{3}+⋯+ℤσ_{n} by (1). Therefore, the central closure \overline{H(E_{ω}, σ)} is a 9-dimensional central special Jordan algebra, which by 2.10 is a division algebra. Then, by 2.11(b), H(E_{ω}, σ) has central degree 3.

We first classify associative tori of central degree 3. Let T = ⊕_{α∈Λ} T_{α} be an associative torus over F of central degree 3 and \overline{T} the central closure over \overline{Z}. By 6.7, we have dim\overline{Z} \overline{T} = 3^{m} for some 2 ≤ m ≤ n. Since \overline{T} is a finite dimensional associative domain, \overline{T} is a division algebra by Wedderburn’s Structure Theorem. Hence m = 2, i.e., dim\overline{Z} \overline{T} = 9. Thus there exists a basis \{σ_{1}, …, σ_{n}\} of Λ such that the central grading group Γ of T is 3ℤσ_{1} + 3ℤσ_{2} + ℤσ_{3} + ⋯ + ℤσ_{n}. Also, it is clear that an associative torus whose central grading group is Γ has central degree 3.

Now, we classify associative tori whose central grading group is Γ. Let 0 ≠ t_{i} ∈ T_{σ_{i}} for 1 ≤ i ≤ n. Then since t_{i}t_{1} = t_{1}t_{i}, t_{i}t_{2} = t_{2}t_{i} and t_{j}t_{i} = t_{i}t_{j} for all 3 ≤ i ≤ n and 1 ≤ j ≤ n, we can identify such a T with the quantum torus F_{q} = F_{q}[t_{1}^{±1}, …, t_{n}^{±1}] determined by q = q(q) where the (1, 2)-entry of q is some q ∈ F^{×}, the (2, 1)-entry is q^{−1} and the other entries are all 1. Moreover, since t_{2}^{3} ∈ Z, we have t_{1}t_{2}^{3} = t_{2}^{3}t_{1} = q^{3}t_{1}t_{2}^{3}, and so q^{3} = 1. If q = 1, then q = q(1) = 1, but the algebra of Laurent polynomials F_{1} cannot have central degree 3 since Z = Z(F_{1}) = F_{1}. Hence q ≠ 1, and F has to contain a primitive 3rd root of unity, say ω. Since q can be either ω or ω^{−1}, let q := q(ω) and q′ := q(ω^{−1}). One can easily see that F_{q} ≅ F_{q′} via t_{1} ↦ t_{2}, t_{2} ↦ t_{1} and t_{i} ↦ t_{i} for i = 3, …, n.

Remark 6.11. Note that F_{q′} can be identified with the opposite algebra F_{q}^{op} of F_{q}. Then F_{q} and F_{q′} are both algebras over their common centre

Z = F [t_{1}^{±3}, t_{2}^{±3}, t_{3}^{±1} …, t_{n}^{±1}].

We showed that F_{q} ≅ F_{q′} over F, but we note that F_{q} ≇ F_{q′} over Z. In general, if A is an associative domain of central degree 3, then we always have A ≇ A^{op} over Z. For, if A ≅ A^{op} over Z, then \overline{A} ≅ \overline{A}^{op} over \overline{Z}, which cannot happen since \overline{A} is a central associative division algebra of degree 3. (See e.g. [19]; if \overline{A} ≅ \overline{A}^{op}, then [\overline{A}]^{2} = 1 in the Brauer group. But the order of \overline{A} in the Brauer group has to divide the degree.)

We summarize the above results as a proposition.

Proposition 6.12. (1) For an associative torus T over F we have: T has central degree 3 ⟺ the central grading group of T is 3ℤσ_{1} + 3ℤσ_{2} + ℤσ_{3} + ⋯ + ℤσ_{n} for some basis {σ_{1}, …, σ_{n}} of Λ.

(2) If ω ∈ F, then an associative torus over F of central degree 3 exists, and any such torus is isomorphic to the quantum torus F_{ω}. If ω ∉ F, e.g. ch. F = 3, no such torus exists. □

Unlike the other types, we will show that the Albert torus 𝔸_{t} is the only Jordan torus of Albert type. We begin with the following:

Proposition 6.13. Let J be a special Jordan torus over F of central degree 3. Then ch. F ≠ 3, and

J ≅_{Λ} \{ \begin{matrix}F_{ω}^{+} & if ω ∈ F \\ H(E_{ω}, σ) & otherwise \end{matrix}

where E = F (\sqrt{−3}). Conversely, the algebras F_{ω}^{+} and H(E_{ω}, σ) are special Jordan tori of central degree 3.

Remark. If ch. F ≠ 3, there exists a primitive 3rd root of unity ω in some extension field of F. Note that F (ω) = F (\sqrt{−3}).

Proof. Only the 1st part remains to be proven. We know that a special Jordan torus J is either a Hermitian torus or a Clifford torus. Since a Clifford torus does not have central degree 3, special Jordan tori of central degree 3 have to be Hermitian tori, i.e., J ≅_{Λ} F_{q}^{+}, H(F_{ε}, ∗) or H(E_{ξ}, σ) (see 4.3 and 4.11). Let Z = Z(J) be the centre of J. By 6.7, there exists a basis {σ_{1}, …, σ_{n}} of Λ such that the central grading group Γ of J has the form

Γ = 3ℤσ_{1} + ⋯ + 3ℤσ_{m} + ℤσ_{m+1} + ⋯ + ℤσ_{n}

for some 2 ≤ m ≤ n and supp J = Λ. In particular, there exists a homogeneous element t ∈ J such that t^{2} ∉ Z (for example, take t to be a nonzero element of degree σ_{1}). If φ : J ˜ ⟶H(F_{ε}, ∗) is a graded isomorphism, then by 3.6(ii), φ(t) is homogeneous in H(F_{ε}, ∗), and φ(t)^{2} ∉ φ(Z) = Z(H(F_{ε}, ∗)). However, for any homogeneous element x ∈ H(F_{ε}, ∗), x^{2} ∈ Z(H(F_{ε}, ∗)) since the entries of ε are ±1. Thus we get a contradiction. Hence the only possible candidates of central degree 3 are F_{q}^{+} and H(E_{ξ}, σ). Since the central closure \overline{J} is a finite dimensional central special Jordan division algebra over \overline{Z} of degree 3, we have, by 2.11(b), dim\overline{Z} \overline{J} = 9. So m has to be 2, by 3.9(iii).

If J ≅_{Λ} F_{q}^{+}, then by 2.5, Z = Z(F_{q}^{+}) = Z(F_{q}), and hence the central grading group of F_{q} is Γ. Thus, by 6.12, we have ω ∈ F, and so ch. F ≠ 3, and q = ω, i.e., J ≅_{Λ} F_{ω}^{+}.

Suppose that J ≅_{Λ} H(E_{ξ}, σ). We identify them. So we have J_{E} = E ⊗_{F} J = E_{ξ}^{+}. Hence, by 2.5 and 3.8, the grading group of Z(E_{ξ}^{+}) = Z(E_{ξ}) is Γ. Then, by 6.12, we get ch. E ≠ 3, ω ∈ E and ξ = ω, i.e., J = H(E_{ω}, σ). Since ωσ(ω) = 1, we have σ(ω) ≠ ω, and so ω ∉ F. Since [E : F] = 2, we obtain E = F (ω) = F (\sqrt{−3}). □

By Zelmanov’s Prime Structure Theorem, if J is a strongly prime exceptional Jordan algebra J over F, then J has central degree 3 and the central closure \overline{J} is an Albert algebra. Let Tr be the generic trace of \overline{J} over \overline{Z}. For a subalgebra U of J, let

U^{⊥} := \{x ∈ J | Tr(Ux) = 0\} ⊂ J ⊂ \overline{J} .

Note that the central closure of an exceptional Jordan domain is an Albert division algebra. The following lemma for an exceptional Jordan domain serves as preparation for the classification of Jordan tori of Albert type. We will show that such a torus satisfies all the assumptions of the lemma.

Lemma 6.14. Let J be an exceptional Jordan domain over F, Z = Z(J) the centre of J and Tr the generic trace of \overline{J}. Let U be a subdomain of J and Z(U) the centre of U. We assume the following conditions:

(i) Z = Z(U) and U has central degree 3,

(ii) U = A^{+} for some associative algebra A over F,

(iii) Tr(U) ⊂ Z,

(iv) there exists an element x ∈ U^{⊥} such that x^{2} ∈ U^{⊥} and z := x^{3} ∈ Z is invertible.

Then, J contains a subalgebra J′ so that there exists

a Z-isomorphism φ : (A, z) ˜ ⟶J′ with φ |_{A}= id and φ((0, 1, 0)) = x or, a Z-isomorphism ψ : (A, z^{−1}) ˜ ⟶J′ with ψ |_{A}= id and ψ((0, 0, 1)) = x,

where (A, z) and (A, z^{−1}) are defined in 6.5.

Moreover, assume that

(v) there exists an F -isomorphism f from A onto the opposite algebra A^{op} such that f ∘ Tr = Tr ∘f and f(z) = z.

Then there exists an F -isomorphism f̃ : (A, z) ˜ ⟶(A, z^{−1}) with f̃ |_{A}= f and f̃((0, 1, 0)) = (0, 0, 1). In particular, J always contains a subalgebra F -isomorphic to (A, z).

Proof. As mentioned above, the central closure \overline{J} is an Albert division algebra over \overline{Z}. By (i), \overline{U} = \overline{Z(U)} ⊗_{Z(U)} U = \overline{Z} ⊗_{Z} U ⊂ \overline{J} is a central division subalgebra over \overline{Z} of degree 3, and tr := Tr |\overline{U} is the generic trace of \overline{U}. By (ii) and 2.5, we have

\overline{U} = \overline{Z(A^{+})} ⊗_{Z(A^{+})} A^{+} = \overline{Z(A)} ⊗_{Z(A)} A^{+} = (\overline{Z(A)} ⊗_{Z(A)} A)^{+} = (\overline{A})^{+}.

Hence B := \overline{A} is a central associative division algebra over \overline{Z} of degree 3, and so A has central degree 3 with tr(A) = tr(U) = Tr(U) ⊂ Z by (iii). Note that the generic trace of A coincides with the generic trace tr of U = A^{+} (see [8] p.230).

Now, since \overline{J} contains \overline{U} = B^{+} for the central simple associative algebra B over \overline{Z} of degree 3, we have \overline{J} ≅ (B, μ) over \overline{Z} for some 0 ≠ μ ∈ \overline{Z} (see [8] p.420). Note that this isomorphism is the identity map on B. So we identify \overline{J} with (B, μ). Since \overline{U}^{⊥} = B^{⊥} = (0, B, B) (see e.g. [18] p.349), we have, by (iv), x = (0, u, v) for some u, v ∈ B. By the multiplication rule 6.1 of (B, μ), we have x^{2} = (2\overline{uv}, u′, v′) for some u′, v′ ∈ B. Since x^{2} ∈ U^{⊥}, we have 2\overline{uv} = 0, and so tr(uv)1 = uv. Hence 0 = 2\overline{uv} = 2\overline{tr(uv)1} = 2 tr(uv)1, and so uv = 0. Since B is an associative division algebra, we have u = 0 or v = 0. If u = v = 0, then x = 0 which contradicts the invertibility of x^{3}. Thus we obtain x = (0, u, 0) or x = (0, 0, v) for some nonzero u, v ∈ B. Then, by 6.3, there exists a \overline{Z}-isomorphism Φ from (B, μ) onto (B, z) or a \overline{Z}-isomorphism Ψ from (B, μ) onto (B, z^{−1}) such that Φ |_{A}= id or Ψ |_{A}= id and Φ(x) = (0, 1, 0) or Ψ(x) = (0, 0, 1). So, Φ(J) contains A and (0, 1, 0) in (B, z), or Ψ(J) contains A and (0, 0, 1) in (B, z^{−1}). Hence Φ(J) contains (A, z) or Ψ(J) contains (A, z^{−1}) (see 6.6). Let J′ = Φ^{−1}((A, z)) or J′ = Ψ^{−1}((A, z^{−1})). Then φ := Φ^{−1} |_{(A,z)} and ψ := Ψ^{−1} |_{(A,z^{−1})} are the required Z-isomorphisms, and so we have shown the first statement.

For the second statement, we use the well-known fact that there exists an isomorphism g : (B, z) ˜ ⟶(B^{op}, z^{−1}) over \overline{Z} defined by g((a_{0}, a_{1}, a_{2})) = (a_{0}, a_{2}, a_{1}) for a_{0}, a_{1}, a_{2} ∈ \overline{A} (see [8] p.422, Exercise 2). Thus

h := g |_{(A,z)}: (A, z) ˜ ⟶(A^{op}, z^{−1})

is a Z-isomorphism. We note that A^{op} has central degree 3, and has the same centre as A. Also, the generic trace of A^{op} coincides with the generic trace tr of A. The F - isomorphism f : A ˜ ⟶A^{op} in our assumption (v) satisfies f(z) = z and f ∘tr = f ∘Tr = Tr ∘f = tr ∘f. So one can check that the map f̄: (A, z^{−1}) ⟶ (A^{op}, z^{−1}) defined by f̄((a_{0}, a_{1}, a_{2})) = (f(a_{0}), f(a_{1}), f(a_{2})) is an F -isomorphism. Consequently, we obtain an F -isomorphism f̃:= f̄^{−1} ∘ h : (A, z) ˜ ⟶(A, z^{−1}) with f̃((0, 1, 0)) = (0, 0, 1). For the last statement, the composition map ψ ∘ f̃ gives an F -isomorphism from (A, z) onto J′. □

Remark. In 6.14, if (A, z) ≅ (A, z^{−1}) over Z, then (\overline{A}, z) ≅ (\overline{A}, z^{−1}) over \overline{Z}. Since (\overline{A}, z) is an Albert division algebra, this cannot happen by [17], p.204 (see 6.11). Hence we always have (A, z) ≇ (A, z^{−1}) over Z though it may happen that (A, z) ≅ (A, z^{−1}) over F. For example, this is the case if (A, z) is a Jordan torus of Albert type below.

We start to classify Jordan tori of Albert type. Let J = ⊕_{α∈Λ} J_{α} be a Jordan torus of Albert type, i.e., the central closure \overline{J} is an Albert algebra over \overline{Z}. Recall that an Albert algebra is a 27-dimensional central simple exceptional Jordan algebra of degree 3. By 6.7 and the fact that we have degree 3 and dimension 27, it follows that supp J = Λ, n ≥ 3, and there exists a basis {σ_{1}, …, σ_{n}} of Λ such that the central grading group Γ of J is given as

Γ = 3ℤσ_{1} + 3ℤσ_{2} + 3ℤσ_{3} + ℤσ_{4} + ⋯ + ℤσ_{n}.

Let U = ⊕_{α∈Δ} J_{α}, where

Δ := ℤσ_{1} + ℤσ_{2} + 3ℤσ_{3} + ℤσ_{4} + ⋯ + ℤσ_{n}.

We claim that Z(U) = Z(J). Since U ⊂ J, we have Z(U) ⊃ Z(J) and so the central grading group Δ_{1} of U lies in between Δ and Γ, i.e., Δ > Δ_{1} > Γ. If Γ ≠ Δ_{1} ≠ Δ, we get |Δ/Δ_{1}| = 3 since |Δ/Γ| = 9. Hence the grading group of the central closure \overline{U} is Δ/Δ_{1} = ℤ_{3}. But by 3.9(iv), this cannot happen. Suppose then that Δ_{1} = Δ. Then U is commutative and associative. We show, using the method of Lemma 2 in [8] p.420, that this cannot happen:

Suppose that U is commutative and associative. Since \overline{J} is an Albert division algebra by 3.4.7, the commutative associative subalgebra \overline{Z} ⊗_{Z} U becomes a subfield of \overline{J}, which is 9-dimensional since |Δ/Γ| = 9. But this is impossible. Indeed, take an algebraic closure Ω containing \overline{Z}. Then, in \overline{J}_{Ω}, we have 1 = e_{1} + ⋯ + e_{9} where e_{i} for i = 1, …, 9 are orthogonal idempotents. Hence, by Lemma 1 in [8] p.229, the degree of \overline{J}_{Ω} is ≥ 9, which is a contradiction since the degree of \overline{J}_{Ω} is equal to the degree of \overline{J} which is 3 (see [8] p.223).

Thus we get Δ_{1} = Γ, i.e., Z(U) = Z. In particular, we have \overline{U} = \overline{Z} ⊗_{Z} U ⊂ \overline{J}. Note that \overline{U} is a central subalgebra of the division algebra \overline{J} which is 9-dimensional since |Δ/Γ| = 9. So by the classification of finite dimensional central simple Jordan algebras, \overline{U} is special (see [8] Corollary 2 p.204 and p.207). Hence, by 2.11, \overline{U} has degree 3. So U is a special Jordan torus of central degree 3. Therefore, by 6.13, ch. F ≠ 3 and U can be identified with F_{ω}^{+} if ω ∈ F and with H(E_{ω}, σ) otherwise, where E = F (\sqrt{−3}) = F (ω).

We first consider the case U = F_{ω}^{+}. Let x be an arbitrary nonzero element in J_{σ_{3}}. Let u_{i} be an arbitrary nonzero element in J_{σ_{i}} for i ≠ 3 and u_{3} := x^{3} ∈ J_{3σ_{3}}. Then F_{ω} = F_{ω}[u_{1}^{±1}, …, u_{n}^{±1}] is a Δ-torus with a ⟨σ_{1}, σ_{2}, 3σ_{3}, σ_{4}, …, σ_{n}⟩-grading. Let Tr be the generic trace of \overline{J} and U^{⊥} = \{y ∈ J | Tr(Uy) = 0\}. We claim that x, x^{2} ∈ U^{⊥}. Since Tr is Z-linear and U is a free Z-module with basis {u_{1}^{i}u_{2}^{j} | i, j = 0, 1, 2}, it is enough to show that Tr ((u_{1}^{i}u_{2}^{j})x^{k}) = 0 for all i, j = 0, 1, 2 and k = 1, 2. Since such (u_{1}^{i}u_{2}^{j})x^{k} are all homogeneous and their degrees are not contained in Γ, we get, by 6.7, Tr ((u_{1}^{i}u_{2}^{j})x^{k}) = 0. Hence our claim is settled.

Since x^{3} = u_{3} ∈ Z is invertible and Tr(F_{ω}) ⊂ Z, we have shown the conditions (i)-(iv) for J = J, U = F_{ω}^{+} and z = u_{3} in 6.14. Therefore, by 6.14, J contains a subalgebra J′ so that

Case (I) φ : (F_{ω}, u_{3}) ˜ ⟶ J′ is a Z-isomorphism
with φ |_{F_{ω}}= id and φ((0, 1, 0)) = x, or
Case (II) ψ : (F_{ω}, u_{3}^{−1}) ˜ ⟶ J′ is a Z-isomorphism
with ψ |_{F_{ω}}= id and ψ((0, 0, 1)) = x.

We give a ⟨σ_{1}, …, σ_{n}⟩-grading to (F_{ω}, u_{3}) so that 𝔸_{t} = (F_{ω}, u_{3}) is the Albert torus, i.e., 𝔸_{t} = ⊕_{α∈Λ} F t_{α}, and t_{σ_{i}} = u_{i} for i ≠ 3 and t_{σ_{3}} = (0, 1, 0) (see 6.8(2)).

Case (I): We have φ(t_{σ_{i}}) = u_{i} ∈ J_{σ_{i}} for i ≠ 3 and φ(t_{σ_{3}}) = x ∈ J_{σ_{3}}. Thus one gets the injective Z-homomorphism φ : 𝔸_{t} ⟶ J with φ(F t_{σ_{i}}) = J_{σ_{i}} for all i = 1, …, n. Since 𝔸_{t} is of strong type, we have, for any α = α_{1}σ_{1} + ⋯ + α_{n}σ_{n} ∈ Λ,

0 ≠ φ(u_{1}^{α_{1}} · (u_{2}^{α_{2}} · (t_{σ_{3}}^{α_{3}} · (u_{4}^{α_{4}} ⋯ u_{n}^{α_{n}})...)) ∈ J_{α},

and hence φ is surjective. Therefore, J ≅ 𝔸_{t} over Z and J ≅_{Λ} 𝔸_{t}.

Case (II): We first show that F_{ω} = F_{ω}[u_{1}^{±1}, …, u_{n}^{±1}] satisfies the condition (v) in 6.14. Clearly the F -linear map f from F_{ω} into F_{ω}^{op} defined by

f(u_{1}^{α_{1}}u_{2}^{α_{2}}u_{3}^{α_{3}} ⋯ u_{n}^{α_{n}}) = u_{2}^{α_{1}}u_{1}^{α_{2}}u_{3}^{α_{3}} ⋯ u_{n}^{α_{n}}

(exchange the first two variables and leave alone the remaining variables) for all α = α_{1}σ_{1} + α_{2}σ_{2} + 3α_{3}σ_{3} + α_{4}σ_{4} + ⋯ + α_{n}σ_{n} ∈ Δ is an F -algebra isomorphism (see 6.11). It is also clear that f ∘ Tr = Tr ∘f and f(u_{3}) = u_{3}. Hence the condition (v) in 6.14 is satisfied, and so there exists an F -isomorphism f̃: 𝔸_{t} = (F_{ω}, u_{3}) ˜ ⟶(F_{ω}, u_{3}^{−1}) with f̃ |_{F_{ω}}= f and f̃((0, 1, 0)) = (0, 0, 1). Thus, by 6.14, we get an injective F - homomorphism ψ ∘ f̃ : 𝔸_{t} ⟶ J with ψ ∘ f̃(F t_{σ_{1}}) = J_{σ_{2}}, ψ ∘ f̃(F t_{σ_{2}}) = J_{σ_{1}} and ψ ∘ f̃(F t_{σ_{i}}) = J_{σ_{i}} for all i = 3, …, n. Since 𝔸_{t} is of strong type, we have, for any α = α_{1}σ_{1} + ⋯ + α_{n}σ_{n} ∈ Λ,

0 ≠ ψ ∘ f̃(u_{1}^{α_{2}} · (u_{2}^{α_{1}} · (t_{σ_{3}}^{α_{3}} · (u_{4}^{α_{4}} ⋯ u_{n}^{α_{n}})...)) ∈ J_{α},

and hence ψ∘f̃is surjective. Therefore, we obtain J ≅ 𝔸_{t} over F. Note that ψ∘f̃is not graded for the ⟨σ_{1}, …, σ_{n}⟩-grading of 𝔸_{t}. However, if we give a ⟨σ_{2}, σ_{1}, σ_{3} …, σ_{n}⟩- grading to 𝔸_{t}, ψ ∘ f̃ is graded and so J ≅_{Λ} 𝔸_{t}.

We finally consider the case U = H(E_{ω}, σ) where E = F (ω). Let J_{E} = E ⊗_{F} J be the Jordan torus over E. Let τ := σ_{E} ⊗ id be a σ_{E}-semilinear involution of J_{E} over F where σ_{E} is the nontrivial Galois automorphism of E over F. Then U_{E} = E ⊗_{F} U = E_{ω}^{+} is an E-subalgebra of J_{E}. Also, by 4.9(2), we have τ |_{E_{ω}}= σ since ω ∉ M_{n}(F). In particular, we have

(6.15)  τ(u_{1}u_{2}) = σ(u_{1}u_{2}) = u_{2}u_{1}.

Now, since J is exceptional, so is J_{E}. Hence the Jordan torus J_{E} over E must be of Albert type since the other two types are special. Since J_{E} contains the subalgebra U_{E} = E_{ω}^{+}, we can apply the previous argument for 0 ≠ x ∈ J_{σ_{3}} ⊂ E⊗_{F} J_{σ_{3}}. Precisely, for u_{3} := x^{3}, we have

𝔸̃_{t} := (E_{ω}, u_{3}) be the Albert torus over E,

and there are two cases: for t := (0, 1, 0) ∈ 𝔸̃_{t}, Case (I): l : J_{E} ˜ ⟶𝔸̃_{t} such that l |_{U_{E}}= l |_{E_{ω}}= id and l(x) = t. Case (II): l′ : J_{E} ˜ ⟶𝔸̃_{t} such that l′(x) = t, l′(u_{1}) = u_{2}, l′(u_{2}) = u_{1} and l′ |_{U_{E}}=

l′ |_{E_{ω}} is an automorphism of the associative algebra E_{ω}.

Case (I): Let τ̃ be the induced involution of 𝔸̃_{t} from τ via the isomorphism between J_{E} and 𝔸̃_{t}, namely, τ̃ = l ∘ τ ∘ l^{−1}. Then we have τ̃(u_{i}) = u_{i} for i = 1, …, n and τ̃(t) = t. Moreover, by 6.15, we have τ̃(u_{1}u_{2}) = u_{2}u_{1}. Let tr be the generic trace of \overline{E}_{ω}. Since tr(u_{1}) = tr(u_{2}) = tr(u_{1}u_{2}) = 0, we have \overline{u_{1}} = −\frac{1}{2}u_{1}, \overline{u_{2}} = −\frac{1}{2}u_{2} and \overline{u_{1}u_{2}} = −\frac{1}{2}u_{1}u_{2}. Hence, by 6.1,

(∗)  (u_{1}u_{2}) · t = (u_{1}u_{2}) · (0, 1, 0) = (0, −\frac{1}{2}u_{1}u_{2}, 0)
= u_{1} · (0, u_{2}, 0) = −2u_{1} · (u_{2} · t).

On the other hand,

(∗∗)  (ωu_{1}u_{2}) · t = (u_{2}u_{1}) · t = τ̃(u_{1}u_{2}) · τ̃(t)
= τ̃((u_{1}u_{2}) · t) = −2τ̃(u_{1} · (u_{2} · t)) by (∗)
= −2τ̃(u_{1}) · (τ̃(u_{2}) · τ̃(t)) = −2u_{1} · (u_{2} · t),

and so (u_{1}u_{2}) · t = (ωu_{1}u_{2}) · t = ω(u_{1}u_{2}) · t, which is absurd since (u_{1}u_{2}) · t ≠ 0.

Case (II): Similarly, let τ̃ := l′ ∘ τ ∘ l′^{−1} be the induced involution of 𝔸̃_{t}. Then we have τ̃(u_{1}) = u_{1}, τ̃(u_{2}) = u_{2}, τ̃(t) = t and τ̃(u_{1}u_{2}) = u_{2}u_{1}. Thus (∗) and (∗∗) also hold for this τ̃. So we get a contradiction.

Consequently, the case ω ∉ F cannot happen. Thus we have proven the following:

Theorem 6.16. Let J be a Jordan n-torus of Albert type over F. Then n ≥ 3, ω ∈ F and J ≅_{Λ} 𝔸_{t} for some toral grading. Conversely, 𝔸_{t} is a Jordan torus of Albert type. □

Combined with Proposition 6.13, we get the following result which is used in [2] Proposition 2.17 p.15 to classify extended affine Lie algebras of type G_{2}:

Corollary 6.17. Let J be a Jordan torus over F of central degree 3. Then ch. F ≠ 3 and

J ≅_{Λ} \{ \begin{matrix}F_{ω}^{+} or 𝔸_{t} & if ω ∈ F \\ H(E_{ω}, σ) & otherwise \end{matrix}

where E = F (ω). Conversely, the algebras F_{ω}^{+}, H(E_{ω}, σ) and 𝔸_{t} are Jordan tori of central degree 3. □

§7 Summary

By 4.11, 5.5 and 6.16, we complete the classification of Jordan tori:

Theorem 7.1. Let J be a Jordan n-torus over F. Then J is graded isomorphic to one of the four special Jordan tori

F_{q}^{+}, H(F_{ε}, ∗), H(E_{ξ}, σ) and J_{S^{(m)}}({a_{ϵ}}_{ϵ∈I}),

or to the Albert torus 𝔸_{t} if n ≥ 3 and F contains a primitive 3rd root of unity. □

Also, by 5.6, we have the following:

Corollary 7.2. Let J be a Jordan n-torus over an algebraically closed field F. Then J is graded isomorphic to one of the three special Jordan tori

F_{q}^{+}, H(F_{ε}, ∗) and a standard Clifford torus J_{S^{(m)}},

or to the Albert torus 𝔸_{t} if n ≥ 3 and ch. F ≠ 3. □

Remark. Martinez and Zelmanov classified strongly prime ℤ-graded Jordan algebras of a certain type in [12]. Our Jordan tori are strongly prime ℤ^{n}-graded Jordan algebras of a very special type. The intersection of their algebras and Jordan tori consists of Jordan 1-tori, which are isomorphic to the algebra of Laurent polynomials F [t, t^{−1}].

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